[{"data":1,"prerenderedAt":4898},["ShallowReactive",2],{"subject-categories-math":3,"formula-de\u002Fformulas\u002Fmath\u002Flinear-algebra\u002Fkugelgleichung":546,"formula-related-de\u002Fformulas\u002Fmath\u002Flinear-algebra\u002Fkugelgleichung":4885},[4,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,102,103,104,105,106,107,108,109,110,111,112,113,114,115,116,117,118,119,120,121,122,123,124,125,126,127,128,129,130,131,132,133,134,135,136,137,138,139,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,164,165,166,167,168,169,170,171,173,174,175,176,177,178,179,180,181,182,183,184,185,186,188,189,190,191,192,193,194,195,196,197,198,199,200,201,202,203,204,205,206,207,209,210,211,212,213,214,215,216,217,218,219,220,221,222,223,224,225,226,227,228,229,230,231,232,233,234,235,236,237,238,239,240,241,242,243,244,245,246,248,249,250,251,252,253,254,255,256,257,258,259,260,261,262,263,264,265,266,267,268,269,270,271,272,273,274,275,276,277,278,279,280,281,282,283,284,285,286,287,288,289,290,291,292,293,294,295,296,297,298,299,300,301,302,303,304,305,306,307,308,309,310,311,313,314,315,316,317,318,319,320,321,322,323,324,325,326,327,328,329,330,331,332,333,334,335,336,337,338,339,340,341,342,343,344,345,346,347,348,349,350,351,352,353,354,355,356,357,358,359,360,361,362,363,364,365,366,367,369,370,371,372,373,374,375,376,377,378,379,380,381,382,383,384,385,386,387,388,389,390,392,393,394,395,396,397,398,399,400,401,402,403,404,405,406,407,408,409,410,411,413,414,415,416,417,418,419,420,421,422,423,424,425,426,427,428,429,430,431,432,433,434,435,436,437,438,439,440,442,443,444,445,446,447,448,449,450,451,452,453,454,455,456,457,458,459,460,461,462,463,464,465,466,467,468,469,470,471,472,473,474,475,476,477,478,479,480,481,482,483,484,485,486,487,488,489,490,491,492,493,494,495,496,497,498,499,500,501,502,503,504,505,506,507,508,509,510,511,512,513,514,515,516,517,519,520,521,522,523,524,525,526,527,528,529,530,531,532,533,534,535,536,537,538,539,540,541,542,543,544,545],{"category":5},"algebra",{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":5},{"category":65},"arithmetic",{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":65},{"category":101},"calculus",{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":101},{"category":172},"differential-equations",{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":172},{"category":187},"discrete-math",{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":187},{"category":208},"functions",{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":208},{"category":247},"geometry",{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":247},{"category":312},"linear-algebra",{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":312},{"category":368},"multivariable-calculus",{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":368},{"category":391},"numerics",{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":391},{"category":412},"sequences-series",{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":412},{"category":441},"stochastics",{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":441},{"category":518},"trigonometry",{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"category":518},{"id":547,"title":548,"aliases":549,"body":556,"category":312,"checks":4831,"conditions":4846,"conditionsHtml":4847,"deprecated":4848,"description":4825,"extension":4849,"figure":4848,"figureCaption":4848,"figureHtml":4848,"formulaId":4850,"latex":4851,"latexHtml":4852,"level":4853,"meta":4854,"navigation":4855,"path":4856,"redirectFrom":4857,"related":4858,"seo":4863,"stem":4864,"subject":583,"summary":4865,"tags":4866,"variables":4872,"__hash__":4884},"formulas_de\u002Fde\u002Fformulas\u002Fmath\u002Flinear-algebra\u002Fkugelgleichung.md","Kugelgleichung",[550,551,552,553,554,555],"Kugelgleichung aufstellen","Kugel Koordinatengleichung","Mittelpunkt und Radius einer Kugel","Punkt auf Kugel prüfen","Kugel Vektorform","Kugelgleichung quadratische Ergänzung",{"type":557,"value":558,"toc":4824},"minimark",[559,564,688,1212,1215,1463,1470,1474,1627,2044,2746,3239,3734,4373,4376,4380],[560,561,563],"h2",{"id":562},"erklärung","Erklärung",[565,566,567,568,572,573,620,621,624,625,655,656,687],"p",{},"Eine ",[569,570,571],"strong",{},"Kugel"," ist die Menge aller Punkte ",[574,575,578,600],"span",{"className":576},[577],"katex",[574,579,582],{"className":580},[581],"katex-mathml",[583,584,586],"math",{"xmlns":585},"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML",[587,588,589,596],"semantics",{},[590,591,592],"mrow",{},[593,594,595],"mi",{},"X",[597,598,595],"annotation",{"encoding":599},"application\u002Fx-tex",[574,601,605],{"className":602,"ariaHidden":604},[603],"katex-html","true",[574,606,609,614],{"className":607},[608],"base",[574,610],{"className":611,"style":613},[612],"strut","height:0.6833em;",[574,615,595],{"className":616,"style":619},[617,618],"mord","mathnormal","margin-right:0.0785em;",", die vom ",[569,622,623],{},"Mittelpunkt"," ",[574,626,628,642],{"className":627},[577],[574,629,631],{"className":630},[581],[583,632,633],{"xmlns":585},[587,634,635,640],{},[590,636,637],{},[593,638,639],{},"M",[597,641,639],{"encoding":599},[574,643,645],{"className":644,"ariaHidden":604},[603],[574,646,648,651],{"className":647},[608],[574,649],{"className":650,"style":613},[612],[574,652,639],{"className":653,"style":654},[617,618],"margin-right:0.109em;"," den festen Abstand ",[574,657,659,673],{"className":658},[577],[574,660,662],{"className":661},[581],[583,663,664],{"xmlns":585},[587,665,666,671],{},[590,667,668],{},[593,669,670],{},"r",[597,672,670],{"encoding":599},[574,674,676],{"className":675,"ariaHidden":604},[603],[574,677,679,683],{"className":678},[608],[574,680],{"className":681,"style":682},[612],"height:0.4306em;",[574,684,670],{"className":685,"style":686},[617,618],"margin-right:0.0278em;"," haben. Mit der Abstandsformel im Raum und Quadrieren ergibt sich die Kugelgleichung. In Vektorform lautet sie",[574,689,692],{"className":690},[691],"katex-display",[574,693,695,802],{"className":694},[577],[574,696,698],{"className":697},[581],[583,699,701],{"xmlns":585,"display":700},"block",[587,702,703,799],{},[590,704,705,709,719,722,729,731,734,736,740,744,746,750,756,758,764,767,770,772,778,780,786,788,790],{},[593,706,708],{"mathvariant":707},"normal","∣",[710,711,712,715],"mover",{"accent":604},[593,713,714],{},"x",[716,717,718],"mo",{},"⃗",[716,720,721],{},"−",[710,723,724,727],{"accent":604},[593,725,726],{},"m",[716,728,718],{},[593,730,708],{"mathvariant":707},[716,732,733],{},"=",[593,735,670],{},[737,738],"mspace",{"width":739},"1em",[741,742,743],"mtext",{},"oder",[737,745],{"width":739},[716,747,749],{"stretchy":748},"false","(",[710,751,752,754],{"accent":604},[593,753,714],{},[716,755,718],{},[716,757,721],{},[710,759,760,762],{"accent":604},[593,761,726],{},[716,763,718],{},[716,765,766],{"stretchy":748},")",[716,768,769],{},"⋅",[716,771,749],{"stretchy":748},[710,773,774,776],{"accent":604},[593,775,714],{},[716,777,718],{},[716,779,721],{},[710,781,782,784],{"accent":604},[593,783,726],{},[716,785,718],{},[716,787,766],{"stretchy":748},[716,789,733],{},[791,792,793,795],"msup",{},[593,794,670],{},[796,797,798],"mn",{},"2",[597,800,801],{"encoding":599},"|\\vec{x} - \\vec{m}| = r \\quad\\text{oder}\\quad (\\vec{x} - \\vec{m}) \\cdot (\\vec{x} - \\vec{m}) = r^2",[574,803,805,882,938,1009,1063,1116,1169],{"className":804,"ariaHidden":604},[603],[574,806,808,812,815,871,875,879],{"className":807},[608],[574,809],{"className":810,"style":811},[612],"height:1em;vertical-align:-0.25em;",[574,813,708],{"className":814},[617],[574,816,819],{"className":817},[617,818],"accent",[574,820,823],{"className":821},[822],"vlist-t",[574,824,827],{"className":825},[826],"vlist-r",[574,828,832,843],{"className":829,"style":831},[830],"vlist","height:0.714em;",[574,833,835,840],{"style":834},"top:-3em;",[574,836],{"className":837,"style":839},[838],"pstrut","height:3em;",[574,841,714],{"className":842},[617,618],[574,844,845,848],{"style":834},[574,846],{"className":847,"style":839},[838],[574,849,853],{"className":850,"style":852},[851],"accent-body","left:-0.2077em;",[574,854,858],{"className":855,"style":857},[856],"overlay","height:0.714em;width:0.471em;",[859,860,867],"svg",{"xmlns":861,"width":862,"height":863,"style":864,"viewBox":865,"preserveAspectRatio":866},"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg","0.471em","0.714em","width:0.471em","0 0 471 714","xMinYMin",[868,869],"path",{"d":870},"M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z",[574,872],{"className":873,"style":874},[737],"margin-right:0.2222em;",[574,876,721],{"className":877},[878],"mbin",[574,880],{"className":881,"style":874},[737],[574,883,885,888,924,927,931,935],{"className":884},[608],[574,886],{"className":887,"style":811},[612],[574,889,891],{"className":890},[617,818],[574,892,894],{"className":893},[822],[574,895,897],{"className":896},[826],[574,898,900,908],{"className":899,"style":831},[830],[574,901,902,905],{"style":834},[574,903],{"className":904,"style":839},[838],[574,906,726],{"className":907},[617,618],[574,909,910,913],{"style":834},[574,911],{"className":912,"style":839},[838],[574,914,917],{"className":915,"style":916},[851],"left:-0.2355em;",[574,918,920],{"className":919,"style":857},[856],[859,921,922],{"xmlns":861,"width":862,"height":863,"style":864,"viewBox":865,"preserveAspectRatio":866},[868,923],{"d":870},[574,925,708],{"className":926},[617],[574,928],{"className":929,"style":930},[737],"margin-right:0.2778em;",[574,932,733],{"className":933},[934],"mrel",[574,936],{"className":937,"style":930},[737],[574,939,941,944,947,951,958,961,965,1000,1003,1006],{"className":940},[608],[574,942],{"className":943,"style":811},[612],[574,945,670],{"className":946,"style":686},[617,618],[574,948],{"className":949,"style":950},[737],"margin-right:1em;",[574,952,955],{"className":953},[617,954],"text",[574,956,743],{"className":957},[617],[574,959],{"className":960,"style":950},[737],[574,962,749],{"className":963},[964],"mopen",[574,966,968],{"className":967},[617,818],[574,969,971],{"className":970},[822],[574,972,974],{"className":973},[826],[574,975,977,985],{"className":976,"style":831},[830],[574,978,979,982],{"style":834},[574,980],{"className":981,"style":839},[838],[574,983,714],{"className":984},[617,618],[574,986,987,990],{"style":834},[574,988],{"className":989,"style":839},[838],[574,991,993],{"className":992,"style":852},[851],[574,994,996],{"className":995,"style":857},[856],[859,997,998],{"xmlns":861,"width":862,"height":863,"style":864,"viewBox":865,"preserveAspectRatio":866},[868,999],{"d":870},[574,1001],{"className":1002,"style":874},[737],[574,1004,721],{"className":1005},[878],[574,1007],{"className":1008,"style":874},[737],[574,1010,1012,1015,1050,1054,1057,1060],{"className":1011},[608],[574,1013],{"className":1014,"style":811},[612],[574,1016,1018],{"className":1017},[617,818],[574,1019,1021],{"className":1020},[822],[574,1022,1024],{"className":1023},[826],[574,1025,1027,1035],{"className":1026,"style":831},[830],[574,1028,1029,1032],{"style":834},[574,1030],{"className":1031,"style":839},[838],[574,1033,726],{"className":1034},[617,618],[574,1036,1037,1040],{"style":834},[574,1038],{"className":1039,"style":839},[838],[574,1041,1043],{"className":1042,"style":916},[851],[574,1044,1046],{"className":1045,"style":857},[856],[859,1047,1048],{"xmlns":861,"width":862,"height":863,"style":864,"viewBox":865,"preserveAspectRatio":866},[868,1049],{"d":870},[574,1051,766],{"className":1052},[1053],"mclose",[574,1055],{"className":1056,"style":874},[737],[574,1058,769],{"className":1059},[878],[574,1061],{"className":1062,"style":874},[737],[574,1064,1066,1069,1072,1107,1110,1113],{"className":1065},[608],[574,1067],{"className":1068,"style":811},[612],[574,1070,749],{"className":1071},[964],[574,1073,1075],{"className":1074},[617,818],[574,1076,1078],{"className":1077},[822],[574,1079,1081],{"className":1080},[826],[574,1082,1084,1092],{"className":1083,"style":831},[830],[574,1085,1086,1089],{"style":834},[574,1087],{"className":1088,"style":839},[838],[574,1090,714],{"className":1091},[617,618],[574,1093,1094,1097],{"style":834},[574,1095],{"className":1096,"style":839},[838],[574,1098,1100],{"className":1099,"style":852},[851],[574,1101,1103],{"className":1102,"style":857},[856],[859,1104,1105],{"xmlns":861,"width":862,"height":863,"style":864,"viewBox":865,"preserveAspectRatio":866},[868,1106],{"d":870},[574,1108],{"className":1109,"style":874},[737],[574,1111,721],{"className":1112},[878],[574,1114],{"className":1115,"style":874},[737],[574,1117,1119,1122,1157,1160,1163,1166],{"className":1118},[608],[574,1120],{"className":1121,"style":811},[612],[574,1123,1125],{"className":1124},[617,818],[574,1126,1128],{"className":1127},[822],[574,1129,1131],{"className":1130},[826],[574,1132,1134,1142],{"className":1133,"style":831},[830],[574,1135,1136,1139],{"style":834},[574,1137],{"className":1138,"style":839},[838],[574,1140,726],{"className":1141},[617,618],[574,1143,1144,1147],{"style":834},[574,1145],{"className":1146,"style":839},[838],[574,1148,1150],{"className":1149,"style":916},[851],[574,1151,1153],{"className":1152,"style":857},[856],[859,1154,1155],{"xmlns":861,"width":862,"height":863,"style":864,"viewBox":865,"preserveAspectRatio":866},[868,1156],{"d":870},[574,1158,766],{"className":1159},[1053],[574,1161],{"className":1162,"style":930},[737],[574,1164,733],{"className":1165},[934],[574,1167],{"className":1168,"style":930},[737],[574,1170,1172,1176],{"className":1171},[608],[574,1173],{"className":1174,"style":1175},[612],"height:0.8641em;",[574,1177,1179,1182],{"className":1178},[617],[574,1180,670],{"className":1181,"style":686},[617,618],[574,1183,1186],{"className":1184},[1185],"msupsub",[574,1187,1189],{"className":1188},[822],[574,1190,1192],{"className":1191},[826],[574,1193,1195],{"className":1194,"style":1175},[830],[574,1196,1198,1202],{"style":1197},"top:-3.113em;margin-right:0.05em;",[574,1199],{"className":1200,"style":1201},[838],"height:2.7em;",[574,1203,1209],{"className":1204},[1205,1206,1207,1208],"sizing","reset-size6","size3","mtight",[574,1210,798],{"className":1211},[617,1208],[565,1213,1214],{},"Sie ist das räumliche Gegenstück zur Kreisgleichung. Setzt man einen Punkt ein, gilt:",[1216,1217,1218,1303,1384],"ul",{},[1219,1220,1221,1222,1298,1299,1302],"li",{},"linke Seite ",[574,1223,1225,1245],{"className":1224},[577],[574,1226,1228],{"className":1227},[581],[583,1229,1230],{"xmlns":585},[587,1231,1232,1242],{},[590,1233,1234,1236],{},[716,1235,733],{},[791,1237,1238,1240],{},[593,1239,670],{},[796,1241,798],{},[597,1243,1244],{"encoding":599},"= r^2",[574,1246,1248,1261],{"className":1247,"ariaHidden":604},[603],[574,1249,1251,1255,1258],{"className":1250},[608],[574,1252],{"className":1253,"style":1254},[612],"height:0.3669em;",[574,1256,733],{"className":1257},[934],[574,1259],{"className":1260,"style":930},[737],[574,1262,1264,1268],{"className":1263},[608],[574,1265],{"className":1266,"style":1267},[612],"height:0.8141em;",[574,1269,1271,1274],{"className":1270},[617],[574,1272,670],{"className":1273,"style":686},[617,618],[574,1275,1277],{"className":1276},[1185],[574,1278,1280],{"className":1279},[822],[574,1281,1283],{"className":1282},[826],[574,1284,1286],{"className":1285,"style":1267},[830],[574,1287,1289,1292],{"style":1288},"top:-3.063em;margin-right:0.05em;",[574,1290],{"className":1291,"style":1201},[838],[574,1293,1295],{"className":1294},[1205,1206,1207,1208],[574,1296,798],{"className":1297},[617,1208],": Der Punkt liegt ",[569,1300,1301],{},"auf"," der Kugel.",[1219,1304,1221,1305,1298,1380,1383],{},[574,1306,1308,1329],{"className":1307},[577],[574,1309,1311],{"className":1310},[581],[583,1312,1313],{"xmlns":585},[587,1314,1315,1326],{},[590,1316,1317,1320],{},[716,1318,1319],{},"\u003C",[791,1321,1322,1324],{},[593,1323,670],{},[796,1325,798],{},[597,1327,1328],{"encoding":599},"\u003C r^2",[574,1330,1332,1345],{"className":1331,"ariaHidden":604},[603],[574,1333,1335,1339,1342],{"className":1334},[608],[574,1336],{"className":1337,"style":1338},[612],"height:0.5782em;vertical-align:-0.0391em;",[574,1340,1319],{"className":1341},[934],[574,1343],{"className":1344,"style":930},[737],[574,1346,1348,1351],{"className":1347},[608],[574,1349],{"className":1350,"style":1267},[612],[574,1352,1354,1357],{"className":1353},[617],[574,1355,670],{"className":1356,"style":686},[617,618],[574,1358,1360],{"className":1359},[1185],[574,1361,1363],{"className":1362},[822],[574,1364,1366],{"className":1365},[826],[574,1367,1369],{"className":1368,"style":1267},[830],[574,1370,1371,1374],{"style":1288},[574,1372],{"className":1373,"style":1201},[838],[574,1375,1377],{"className":1376},[1205,1206,1207,1208],[574,1378,798],{"className":1379},[617,1208],[569,1381,1382],{},"innerhalb",".",[1219,1385,1221,1386,1298,1460,1383],{},[574,1387,1389,1410],{"className":1388},[577],[574,1390,1392],{"className":1391},[581],[583,1393,1394],{"xmlns":585},[587,1395,1396,1407],{},[590,1397,1398,1401],{},[716,1399,1400],{},">",[791,1402,1403,1405],{},[593,1404,670],{},[796,1406,798],{},[597,1408,1409],{"encoding":599},"> r^2",[574,1411,1413,1425],{"className":1412,"ariaHidden":604},[603],[574,1414,1416,1419,1422],{"className":1415},[608],[574,1417],{"className":1418,"style":1338},[612],[574,1420,1400],{"className":1421},[934],[574,1423],{"className":1424,"style":930},[737],[574,1426,1428,1431],{"className":1427},[608],[574,1429],{"className":1430,"style":1267},[612],[574,1432,1434,1437],{"className":1433},[617],[574,1435,670],{"className":1436,"style":686},[617,618],[574,1438,1440],{"className":1439},[1185],[574,1441,1443],{"className":1442},[822],[574,1444,1446],{"className":1445},[826],[574,1447,1449],{"className":1448,"style":1267},[830],[574,1450,1451,1454],{"style":1288},[574,1452],{"className":1453,"style":1201},[838],[574,1455,1457],{"className":1456},[1205,1206,1207,1208],[574,1458,798],{"className":1459},[617,1208],[569,1461,1462],{},"außerhalb",[565,1464,1465,1466,1469],{},"Ist die Gleichung ausmultipliziert gegeben, findet man Mittelpunkt und Radius mit ",[569,1467,1468],{},"quadratischer Ergänzung"," für jede Koordinate.",[560,1471,1473],{"id":1472},"beispiele","Beispiele",[565,1475,1476,1477,1572,1573,1626],{},"Kugel mit ",[574,1478,1480,1512],{"className":1479},[577],[574,1481,1483],{"className":1482},[581],[583,1484,1485],{"xmlns":585},[587,1486,1487,1509],{},[590,1488,1489,1491,1493,1496,1498,1500,1502,1504,1507],{},[593,1490,639],{},[716,1492,749],{"stretchy":748},[796,1494,1495],{},"1",[716,1497,708],{},[716,1499,721],{},[796,1501,798],{},[716,1503,708],{},[796,1505,1506],{},"3",[716,1508,766],{"stretchy":748},[597,1510,1511],{"encoding":599},"M(1 \\mid -2 \\mid 3)",[574,1513,1515,1539,1560],{"className":1514,"ariaHidden":604},[603],[574,1516,1518,1521,1524,1527,1530,1533,1536],{"className":1517},[608],[574,1519],{"className":1520,"style":811},[612],[574,1522,639],{"className":1523,"style":654},[617,618],[574,1525,749],{"className":1526},[964],[574,1528,1495],{"className":1529},[617],[574,1531],{"className":1532,"style":930},[737],[574,1534,708],{"className":1535},[934],[574,1537],{"className":1538,"style":930},[737],[574,1540,1542,1545,1548,1551,1554,1557],{"className":1541},[608],[574,1543],{"className":1544,"style":811},[612],[574,1546,721],{"className":1547},[617],[574,1549,798],{"className":1550},[617],[574,1552],{"className":1553,"style":930},[737],[574,1555,708],{"className":1556},[934],[574,1558],{"className":1559,"style":930},[737],[574,1561,1563,1566,1569],{"className":1562},[608],[574,1564],{"className":1565,"style":811},[612],[574,1567,1506],{"className":1568},[617],[574,1570,766],{"className":1571},[1053]," und ",[574,1574,1576,1595],{"className":1575},[577],[574,1577,1579],{"className":1578},[581],[583,1580,1581],{"xmlns":585},[587,1582,1583,1592],{},[590,1584,1585,1587,1589],{},[593,1586,670],{},[716,1588,733],{},[796,1590,1591],{},"4",[597,1593,1594],{"encoding":599},"r = 4",[574,1596,1598,1616],{"className":1597,"ariaHidden":604},[603],[574,1599,1601,1604,1607,1610,1613],{"className":1600},[608],[574,1602],{"className":1603,"style":682},[612],[574,1605,670],{"className":1606,"style":686},[617,618],[574,1608],{"className":1609,"style":930},[737],[574,1611,733],{"className":1612},[934],[574,1614],{"className":1615,"style":930},[737],[574,1617,1619,1623],{"className":1618},[608],[574,1620],{"className":1621,"style":1622},[612],"height:0.6444em;",[574,1624,1591],{"className":1625},[617],":",[574,1628,1630],{"className":1629},[691],[574,1631,1633,1710],{"className":1632},[577],[574,1634,1636],{"className":1635},[581],[583,1637,1638],{"xmlns":585,"display":700},[587,1639,1640,1707],{},[590,1641,1642,1644,1651,1653,1655,1661,1664,1666,1672,1674,1676,1682,1684,1686,1692,1694,1696,1702,1704],{},[716,1643,749],{"stretchy":748},[1645,1646,1647,1649],"msub",{},[593,1648,714],{},[796,1650,1495],{},[716,1652,721],{},[796,1654,1495],{},[791,1656,1657,1659],{},[716,1658,766],{"stretchy":748},[796,1660,798],{},[716,1662,1663],{},"+",[716,1665,749],{"stretchy":748},[1645,1667,1668,1670],{},[593,1669,714],{},[796,1671,798],{},[716,1673,1663],{},[796,1675,798],{},[791,1677,1678,1680],{},[716,1679,766],{"stretchy":748},[796,1681,798],{},[716,1683,1663],{},[716,1685,749],{"stretchy":748},[1645,1687,1688,1690],{},[593,1689,714],{},[796,1691,1506],{},[716,1693,721],{},[796,1695,1506],{},[791,1697,1698,1700],{},[716,1699,766],{"stretchy":748},[796,1701,798],{},[716,1703,733],{},[796,1705,1706],{},"16",[597,1708,1709],{"encoding":599},"(x_1 - 1)^2 + (x_2 + 2)^2 + (x_3 - 3)^2 = 16",[574,1711,1713,1777,1825,1883,1930,1988,2035],{"className":1712,"ariaHidden":604},[603],[574,1714,1716,1719,1722,1768,1771,1774],{"className":1715},[608],[574,1717],{"className":1718,"style":811},[612],[574,1720,749],{"className":1721},[964],[574,1723,1725,1728],{"className":1724},[617],[574,1726,714],{"className":1727},[617,618],[574,1729,1731],{"className":1730},[1185],[574,1732,1735,1759],{"className":1733},[822,1734],"vlist-t2",[574,1736,1738,1754],{"className":1737},[826],[574,1739,1742],{"className":1740,"style":1741},[830],"height:0.3011em;",[574,1743,1745,1748],{"style":1744},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[574,1746],{"className":1747,"style":1201},[838],[574,1749,1751],{"className":1750},[1205,1206,1207,1208],[574,1752,1495],{"className":1753},[617,1208],[574,1755,1758],{"className":1756},[1757],"vlist-s","​",[574,1760,1762],{"className":1761},[826],[574,1763,1766],{"className":1764,"style":1765},[830],"height:0.15em;",[574,1767],{},[574,1769],{"className":1770,"style":874},[737],[574,1772,721],{"className":1773},[878],[574,1775],{"className":1776,"style":874},[737],[574,1778,1780,1784,1787,1816,1819,1822],{"className":1779},[608],[574,1781],{"className":1782,"style":1783},[612],"height:1.1141em;vertical-align:-0.25em;",[574,1785,1495],{"className":1786},[617],[574,1788,1790,1793],{"className":1789},[1053],[574,1791,766],{"className":1792},[1053],[574,1794,1796],{"className":1795},[1185],[574,1797,1799],{"className":1798},[822],[574,1800,1802],{"className":1801},[826],[574,1803,1805],{"className":1804,"style":1175},[830],[574,1806,1807,1810],{"style":1197},[574,1808],{"className":1809,"style":1201},[838],[574,1811,1813],{"className":1812},[1205,1206,1207,1208],[574,1814,798],{"className":1815},[617,1208],[574,1817],{"className":1818,"style":874},[737],[574,1820,1663],{"className":1821},[878],[574,1823],{"className":1824,"style":874},[737],[574,1826,1828,1831,1834,1874,1877,1880],{"className":1827},[608],[574,1829],{"className":1830,"style":811},[612],[574,1832,749],{"className":1833},[964],[574,1835,1837,1840],{"className":1836},[617],[574,1838,714],{"className":1839},[617,618],[574,1841,1843],{"className":1842},[1185],[574,1844,1846,1866],{"className":1845},[822,1734],[574,1847,1849,1863],{"className":1848},[826],[574,1850,1852],{"className":1851,"style":1741},[830],[574,1853,1854,1857],{"style":1744},[574,1855],{"className":1856,"style":1201},[838],[574,1858,1860],{"className":1859},[1205,1206,1207,1208],[574,1861,798],{"className":1862},[617,1208],[574,1864,1758],{"className":1865},[1757],[574,1867,1869],{"className":1868},[826],[574,1870,1872],{"className":1871,"style":1765},[830],[574,1873],{},[574,1875],{"className":1876,"style":874},[737],[574,1878,1663],{"className":1879},[878],[574,1881],{"className":1882,"style":874},[737],[574,1884,1886,1889,1892,1921,1924,1927],{"className":1885},[608],[574,1887],{"className":1888,"style":1783},[612],[574,1890,798],{"className":1891},[617],[574,1893,1895,1898],{"className":1894},[1053],[574,1896,766],{"className":1897},[1053],[574,1899,1901],{"className":1900},[1185],[574,1902,1904],{"className":1903},[822],[574,1905,1907],{"className":1906},[826],[574,1908,1910],{"className":1909,"style":1175},[830],[574,1911,1912,1915],{"style":1197},[574,1913],{"className":1914,"style":1201},[838],[574,1916,1918],{"className":1917},[1205,1206,1207,1208],[574,1919,798],{"className":1920},[617,1208],[574,1922],{"className":1923,"style":874},[737],[574,1925,1663],{"className":1926},[878],[574,1928],{"className":1929,"style":874},[737],[574,1931,1933,1936,1939,1979,1982,1985],{"className":1932},[608],[574,1934],{"className":1935,"style":811},[612],[574,1937,749],{"className":1938},[964],[574,1940,1942,1945],{"className":1941},[617],[574,1943,714],{"className":1944},[617,618],[574,1946,1948],{"className":1947},[1185],[574,1949,1951,1971],{"className":1950},[822,1734],[574,1952,1954,1968],{"className":1953},[826],[574,1955,1957],{"className":1956,"style":1741},[830],[574,1958,1959,1962],{"style":1744},[574,1960],{"className":1961,"style":1201},[838],[574,1963,1965],{"className":1964},[1205,1206,1207,1208],[574,1966,1506],{"className":1967},[617,1208],[574,1969,1758],{"className":1970},[1757],[574,1972,1974],{"className":1973},[826],[574,1975,1977],{"className":1976,"style":1765},[830],[574,1978],{},[574,1980],{"className":1981,"style":874},[737],[574,1983,721],{"className":1984},[878],[574,1986],{"className":1987,"style":874},[737],[574,1989,1991,1994,1997,2026,2029,2032],{"className":1990},[608],[574,1992],{"className":1993,"style":1783},[612],[574,1995,1506],{"className":1996},[617],[574,1998,2000,2003],{"className":1999},[1053],[574,2001,766],{"className":2002},[1053],[574,2004,2006],{"className":2005},[1185],[574,2007,2009],{"className":2008},[822],[574,2010,2012],{"className":2011},[826],[574,2013,2015],{"className":2014,"style":1175},[830],[574,2016,2017,2020],{"style":1197},[574,2018],{"className":2019,"style":1201},[838],[574,2021,2023],{"className":2022},[1205,1206,1207,1208],[574,2024,798],{"className":2025},[617,1208],[574,2027],{"className":2028,"style":930},[737],[574,2030,733],{"className":2031},[934],[574,2033],{"className":2034,"style":930},[737],[574,2036,2038,2041],{"className":2037},[608],[574,2039],{"className":2040,"style":1622},[612],[574,2042,1706],{"className":2043},[617],[1216,2045,2046,2266,2506],{},[1219,2047,2048,2138,2139,2236,2237,2265],{},[574,2049,2051,2080],{"className":2050},[577],[574,2052,2054],{"className":2053},[581],[583,2055,2056],{"xmlns":585},[587,2057,2058,2077],{},[590,2059,2060,2063,2065,2067,2069,2071,2073,2075],{},[593,2061,2062],{},"P",[716,2064,749],{"stretchy":748},[796,2066,1495],{},[716,2068,708],{},[796,2070,798],{},[716,2072,708],{},[796,2074,1506],{},[716,2076,766],{"stretchy":748},[597,2078,2079],{"encoding":599},"P(1 \\mid 2 \\mid 3)",[574,2081,2083,2108,2126],{"className":2082,"ariaHidden":604},[603],[574,2084,2086,2089,2093,2096,2099,2102,2105],{"className":2085},[608],[574,2087],{"className":2088,"style":811},[612],[574,2090,2062],{"className":2091,"style":2092},[617,618],"margin-right:0.1389em;",[574,2094,749],{"className":2095},[964],[574,2097,1495],{"className":2098},[617],[574,2100],{"className":2101,"style":930},[737],[574,2103,708],{"className":2104},[934],[574,2106],{"className":2107,"style":930},[737],[574,2109,2111,2114,2117,2120,2123],{"className":2110},[608],[574,2112],{"className":2113,"style":811},[612],[574,2115,798],{"className":2116},[617],[574,2118],{"className":2119,"style":930},[737],[574,2121,708],{"className":2122},[934],[574,2124],{"className":2125,"style":930},[737],[574,2127,2129,2132,2135],{"className":2128},[608],[574,2130],{"className":2131,"style":811},[612],[574,2133,1506],{"className":2134},[617],[574,2136,766],{"className":2137},[1053],": ",[574,2140,2142,2169],{"className":2141},[577],[574,2143,2145],{"className":2144},[581],[583,2146,2147],{"xmlns":585},[587,2148,2149,2166],{},[590,2150,2151,2154,2156,2158,2160,2162,2164],{},[796,2152,2153],{},"0",[716,2155,1663],{},[796,2157,1706],{},[716,2159,1663],{},[796,2161,2153],{},[716,2163,733],{},[796,2165,1706],{},[597,2167,2168],{"encoding":599},"0 + 16 + 0 = 16",[574,2170,2172,2191,2209,2227],{"className":2171,"ariaHidden":604},[603],[574,2173,2175,2179,2182,2185,2188],{"className":2174},[608],[574,2176],{"className":2177,"style":2178},[612],"height:0.7278em;vertical-align:-0.0833em;",[574,2180,2153],{"className":2181},[617],[574,2183],{"className":2184,"style":874},[737],[574,2186,1663],{"className":2187},[878],[574,2189],{"className":2190,"style":874},[737],[574,2192,2194,2197,2200,2203,2206],{"className":2193},[608],[574,2195],{"className":2196,"style":2178},[612],[574,2198,1706],{"className":2199},[617],[574,2201],{"className":2202,"style":874},[737],[574,2204,1663],{"className":2205},[878],[574,2207],{"className":2208,"style":874},[737],[574,2210,2212,2215,2218,2221,2224],{"className":2211},[608],[574,2213],{"className":2214,"style":1622},[612],[574,2216,2153],{"className":2217},[617],[574,2219],{"className":2220,"style":930},[737],[574,2222,733],{"className":2223},[934],[574,2225],{"className":2226,"style":930},[737],[574,2228,2230,2233],{"className":2229},[608],[574,2231],{"className":2232,"style":1622},[612],[574,2234,1706],{"className":2235},[617],", ",[574,2238,2240,2253],{"className":2239},[577],[574,2241,2243],{"className":2242},[581],[583,2244,2245],{"xmlns":585},[587,2246,2247,2251],{},[590,2248,2249],{},[593,2250,2062],{},[597,2252,2062],{"encoding":599},[574,2254,2256],{"className":2255,"ariaHidden":604},[603],[574,2257,2259,2262],{"className":2258},[608],[574,2260],{"className":2261,"style":613},[612],[574,2263,2062],{"className":2264,"style":2092},[617,618]," liegt auf der Kugel.",[1219,2267,2268,2138,2357,2236,2476,2505],{},[574,2269,2271,2300],{"className":2270},[577],[574,2272,2274],{"className":2273},[581],[583,2275,2276],{"xmlns":585},[587,2277,2278,2297],{},[590,2279,2280,2283,2285,2287,2289,2291,2293,2295],{},[593,2281,2282],{},"Q",[716,2284,749],{"stretchy":748},[796,2286,1506],{},[716,2288,708],{},[796,2290,2153],{},[716,2292,708],{},[796,2294,1591],{},[716,2296,766],{"stretchy":748},[597,2298,2299],{"encoding":599},"Q(3 \\mid 0 \\mid 4)",[574,2301,2303,2327,2345],{"className":2302,"ariaHidden":604},[603],[574,2304,2306,2309,2312,2315,2318,2321,2324],{"className":2305},[608],[574,2307],{"className":2308,"style":811},[612],[574,2310,2282],{"className":2311},[617,618],[574,2313,749],{"className":2314},[964],[574,2316,1506],{"className":2317},[617],[574,2319],{"className":2320,"style":930},[737],[574,2322,708],{"className":2323},[934],[574,2325],{"className":2326,"style":930},[737],[574,2328,2330,2333,2336,2339,2342],{"className":2329},[608],[574,2331],{"className":2332,"style":811},[612],[574,2334,2153],{"className":2335},[617],[574,2337],{"className":2338,"style":930},[737],[574,2340,708],{"className":2341},[934],[574,2343],{"className":2344,"style":930},[737],[574,2346,2348,2351,2354],{"className":2347},[608],[574,2349],{"className":2350,"style":811},[612],[574,2352,1591],{"className":2353},[617],[574,2355,766],{"className":2356},[1053],[574,2358,2360,2391],{"className":2359},[577],[574,2361,2363],{"className":2362},[581],[583,2364,2365],{"xmlns":585},[587,2366,2367,2388],{},[590,2368,2369,2371,2373,2375,2377,2379,2381,2384,2386],{},[796,2370,1591],{},[716,2372,1663],{},[796,2374,1591],{},[716,2376,1663],{},[796,2378,1495],{},[716,2380,733],{},[796,2382,2383],{},"9",[716,2385,1319],{},[796,2387,1706],{},[597,2389,2390],{"encoding":599},"4 + 4 + 1 = 9 \u003C 16",[574,2392,2394,2412,2430,2448,2467],{"className":2393,"ariaHidden":604},[603],[574,2395,2397,2400,2403,2406,2409],{"className":2396},[608],[574,2398],{"className":2399,"style":2178},[612],[574,2401,1591],{"className":2402},[617],[574,2404],{"className":2405,"style":874},[737],[574,2407,1663],{"className":2408},[878],[574,2410],{"className":2411,"style":874},[737],[574,2413,2415,2418,2421,2424,2427],{"className":2414},[608],[574,2416],{"className":2417,"style":2178},[612],[574,2419,1591],{"className":2420},[617],[574,2422],{"className":2423,"style":874},[737],[574,2425,1663],{"className":2426},[878],[574,2428],{"className":2429,"style":874},[737],[574,2431,2433,2436,2439,2442,2445],{"className":2432},[608],[574,2434],{"className":2435,"style":1622},[612],[574,2437,1495],{"className":2438},[617],[574,2440],{"className":2441,"style":930},[737],[574,2443,733],{"className":2444},[934],[574,2446],{"className":2447,"style":930},[737],[574,2449,2451,2455,2458,2461,2464],{"className":2450},[608],[574,2452],{"className":2453,"style":2454},[612],"height:0.6835em;vertical-align:-0.0391em;",[574,2456,2383],{"className":2457},[617],[574,2459],{"className":2460,"style":930},[737],[574,2462,1319],{"className":2463},[934],[574,2465],{"className":2466,"style":930},[737],[574,2468,2470,2473],{"className":2469},[608],[574,2471],{"className":2472,"style":1622},[612],[574,2474,1706],{"className":2475},[617],[574,2477,2479,2492],{"className":2478},[577],[574,2480,2482],{"className":2481},[581],[583,2483,2484],{"xmlns":585},[587,2485,2486,2490],{},[590,2487,2488],{},[593,2489,2282],{},[597,2491,2282],{"encoding":599},[574,2493,2495],{"className":2494,"ariaHidden":604},[603],[574,2496,2498,2502],{"className":2497},[608],[574,2499],{"className":2500,"style":2501},[612],"height:0.8778em;vertical-align:-0.1944em;",[574,2503,2282],{"className":2504},[617,618]," liegt innerhalb.",[1219,2507,2508,2138,2599,2236,2717,2745],{},[574,2509,2511,2541],{"className":2510},[577],[574,2512,2514],{"className":2513},[581],[583,2515,2516],{"xmlns":585},[587,2517,2518,2538],{},[590,2519,2520,2523,2525,2528,2530,2532,2534,2536],{},[593,2521,2522],{},"R",[716,2524,749],{"stretchy":748},[796,2526,2527],{},"5",[716,2529,708],{},[796,2531,798],{},[716,2533,708],{},[796,2535,1506],{},[716,2537,766],{"stretchy":748},[597,2539,2540],{"encoding":599},"R(5 \\mid 2 \\mid 3)",[574,2542,2544,2569,2587],{"className":2543,"ariaHidden":604},[603],[574,2545,2547,2550,2554,2557,2560,2563,2566],{"className":2546},[608],[574,2548],{"className":2549,"style":811},[612],[574,2551,2522],{"className":2552,"style":2553},[617,618],"margin-right:0.0077em;",[574,2555,749],{"className":2556},[964],[574,2558,2527],{"className":2559},[617],[574,2561],{"className":2562,"style":930},[737],[574,2564,708],{"className":2565},[934],[574,2567],{"className":2568,"style":930},[737],[574,2570,2572,2575,2578,2581,2584],{"className":2571},[608],[574,2573],{"className":2574,"style":811},[612],[574,2576,798],{"className":2577},[617],[574,2579],{"className":2580,"style":930},[737],[574,2582,708],{"className":2583},[934],[574,2585],{"className":2586,"style":930},[737],[574,2588,2590,2593,2596],{"className":2589},[608],[574,2591],{"className":2592,"style":811},[612],[574,2594,1506],{"className":2595},[617],[574,2597,766],{"className":2598},[1053],[574,2600,2602,2633],{"className":2601},[577],[574,2603,2605],{"className":2604},[581],[583,2606,2607],{"xmlns":585},[587,2608,2609,2630],{},[590,2610,2611,2613,2615,2617,2619,2621,2623,2626,2628],{},[796,2612,1706],{},[716,2614,1663],{},[796,2616,1706],{},[716,2618,1663],{},[796,2620,2153],{},[716,2622,733],{},[796,2624,2625],{},"32",[716,2627,1400],{},[796,2629,1706],{},[597,2631,2632],{"encoding":599},"16 + 16 + 0 = 32 > 16",[574,2634,2636,2654,2672,2690,2708],{"className":2635,"ariaHidden":604},[603],[574,2637,2639,2642,2645,2648,2651],{"className":2638},[608],[574,2640],{"className":2641,"style":2178},[612],[574,2643,1706],{"className":2644},[617],[574,2646],{"className":2647,"style":874},[737],[574,2649,1663],{"className":2650},[878],[574,2652],{"className":2653,"style":874},[737],[574,2655,2657,2660,2663,2666,2669],{"className":2656},[608],[574,2658],{"className":2659,"style":2178},[612],[574,2661,1706],{"className":2662},[617],[574,2664],{"className":2665,"style":874},[737],[574,2667,1663],{"className":2668},[878],[574,2670],{"className":2671,"style":874},[737],[574,2673,2675,2678,2681,2684,2687],{"className":2674},[608],[574,2676],{"className":2677,"style":1622},[612],[574,2679,2153],{"className":2680},[617],[574,2682],{"className":2683,"style":930},[737],[574,2685,733],{"className":2686},[934],[574,2688],{"className":2689,"style":930},[737],[574,2691,2693,2696,2699,2702,2705],{"className":2692},[608],[574,2694],{"className":2695,"style":2454},[612],[574,2697,2625],{"className":2698},[617],[574,2700],{"className":2701,"style":930},[737],[574,2703,1400],{"className":2704},[934],[574,2706],{"className":2707,"style":930},[737],[574,2709,2711,2714],{"className":2710},[608],[574,2712],{"className":2713,"style":1622},[612],[574,2715,1706],{"className":2716},[617],[574,2718,2720,2733],{"className":2719},[577],[574,2721,2723],{"className":2722},[581],[583,2724,2725],{"xmlns":585},[587,2726,2727,2731],{},[590,2728,2729],{},[593,2730,2522],{},[597,2732,2522],{"encoding":599},[574,2734,2736],{"className":2735,"ariaHidden":604},[603],[574,2737,2739,2742],{"className":2738},[608],[574,2740],{"className":2741,"style":613},[612],[574,2743,2522],{"className":2744,"style":2553},[617,618]," liegt außerhalb.",[565,2747,2748,2749,3238],{},"Umgekehrt: Gegeben ist ",[574,2750,2752,2832],{"className":2751},[577],[574,2753,2755],{"className":2754},[581],[583,2756,2757],{"xmlns":585},[587,2758,2759,2829],{},[590,2760,2761,2770,2772,2780,2782,2790,2792,2794,2800,2802,2804,2810,2812,2815,2821,2823,2825,2827],{},[2762,2763,2764,2766,2768],"msubsup",{},[593,2765,714],{},[796,2767,1495],{},[796,2769,798],{},[716,2771,1663],{},[2762,2773,2774,2776,2778],{},[593,2775,714],{},[796,2777,798],{},[796,2779,798],{},[716,2781,1663],{},[2762,2783,2784,2786,2788],{},[593,2785,714],{},[796,2787,1506],{},[796,2789,798],{},[716,2791,721],{},[796,2793,798],{},[1645,2795,2796,2798],{},[593,2797,714],{},[796,2799,1495],{},[716,2801,1663],{},[796,2803,1591],{},[1645,2805,2806,2808],{},[593,2807,714],{},[796,2809,798],{},[716,2811,721],{},[796,2813,2814],{},"6",[1645,2816,2817,2819],{},[593,2818,714],{},[796,2820,1506],{},[716,2822,721],{},[796,2824,798],{},[716,2826,733],{},[796,2828,2153],{},[597,2830,2831],{"encoding":599},"x_1^2 + x_2^2 + x_3^2 - 2x_1 + 4x_2 - 6x_3 - 2 = 0",[574,2833,2835,2904,2970,3036,3095,3153,3211,3229],{"className":2834,"ariaHidden":604},[603],[574,2836,2838,2842,2895,2898,2901],{"className":2837},[608],[574,2839],{"className":2840,"style":2841},[612],"height:1.0622em;vertical-align:-0.2481em;",[574,2843,2845,2848],{"className":2844},[617],[574,2846,714],{"className":2847},[617,618],[574,2849,2851],{"className":2850},[1185],[574,2852,2854,2886],{"className":2853},[822,1734],[574,2855,2857,2883],{"className":2856},[826],[574,2858,2860,2872],{"className":2859,"style":1267},[830],[574,2861,2863,2866],{"style":2862},"top:-2.4519em;margin-left:0em;margin-right:0.05em;",[574,2864],{"className":2865,"style":1201},[838],[574,2867,2869],{"className":2868},[1205,1206,1207,1208],[574,2870,1495],{"className":2871},[617,1208],[574,2873,2874,2877],{"style":1288},[574,2875],{"className":2876,"style":1201},[838],[574,2878,2880],{"className":2879},[1205,1206,1207,1208],[574,2881,798],{"className":2882},[617,1208],[574,2884,1758],{"className":2885},[1757],[574,2887,2889],{"className":2888},[826],[574,2890,2893],{"className":2891,"style":2892},[830],"height:0.2481em;",[574,2894],{},[574,2896],{"className":2897,"style":874},[737],[574,2899,1663],{"className":2900},[878],[574,2902],{"className":2903,"style":874},[737],[574,2905,2907,2910,2961,2964,2967],{"className":2906},[608],[574,2908],{"className":2909,"style":2841},[612],[574,2911,2913,2916],{"className":2912},[617],[574,2914,714],{"className":2915},[617,618],[574,2917,2919],{"className":2918},[1185],[574,2920,2922,2953],{"className":2921},[822,1734],[574,2923,2925,2950],{"className":2924},[826],[574,2926,2928,2939],{"className":2927,"style":1267},[830],[574,2929,2930,2933],{"style":2862},[574,2931],{"className":2932,"style":1201},[838],[574,2934,2936],{"className":2935},[1205,1206,1207,1208],[574,2937,798],{"className":2938},[617,1208],[574,2940,2941,2944],{"style":1288},[574,2942],{"className":2943,"style":1201},[838],[574,2945,2947],{"className":2946},[1205,1206,1207,1208],[574,2948,798],{"className":2949},[617,1208],[574,2951,1758],{"className":2952},[1757],[574,2954,2956],{"className":2955},[826],[574,2957,2959],{"className":2958,"style":2892},[830],[574,2960],{},[574,2962],{"className":2963,"style":874},[737],[574,2965,1663],{"className":2966},[878],[574,2968],{"className":2969,"style":874},[737],[574,2971,2973,2976,3027,3030,3033],{"className":2972},[608],[574,2974],{"className":2975,"style":2841},[612],[574,2977,2979,2982],{"className":2978},[617],[574,2980,714],{"className":2981},[617,618],[574,2983,2985],{"className":2984},[1185],[574,2986,2988,3019],{"className":2987},[822,1734],[574,2989,2991,3016],{"className":2990},[826],[574,2992,2994,3005],{"className":2993,"style":1267},[830],[574,2995,2996,2999],{"style":2862},[574,2997],{"className":2998,"style":1201},[838],[574,3000,3002],{"className":3001},[1205,1206,1207,1208],[574,3003,1506],{"className":3004},[617,1208],[574,3006,3007,3010],{"style":1288},[574,3008],{"className":3009,"style":1201},[838],[574,3011,3013],{"className":3012},[1205,1206,1207,1208],[574,3014,798],{"className":3015},[617,1208],[574,3017,1758],{"className":3018},[1757],[574,3020,3022],{"className":3021},[826],[574,3023,3025],{"className":3024,"style":2892},[830],[574,3026],{},[574,3028],{"className":3029,"style":874},[737],[574,3031,721],{"className":3032},[878],[574,3034],{"className":3035,"style":874},[737],[574,3037,3039,3043,3046,3086,3089,3092],{"className":3038},[608],[574,3040],{"className":3041,"style":3042},[612],"height:0.7944em;vertical-align:-0.15em;",[574,3044,798],{"className":3045},[617],[574,3047,3049,3052],{"className":3048},[617],[574,3050,714],{"className":3051},[617,618],[574,3053,3055],{"className":3054},[1185],[574,3056,3058,3078],{"className":3057},[822,1734],[574,3059,3061,3075],{"className":3060},[826],[574,3062,3064],{"className":3063,"style":1741},[830],[574,3065,3066,3069],{"style":1744},[574,3067],{"className":3068,"style":1201},[838],[574,3070,3072],{"className":3071},[1205,1206,1207,1208],[574,3073,1495],{"className":3074},[617,1208],[574,3076,1758],{"className":3077},[1757],[574,3079,3081],{"className":3080},[826],[574,3082,3084],{"className":3083,"style":1765},[830],[574,3085],{},[574,3087],{"className":3088,"style":874},[737],[574,3090,1663],{"className":3091},[878],[574,3093],{"className":3094,"style":874},[737],[574,3096,3098,3101,3104,3144,3147,3150],{"className":3097},[608],[574,3099],{"className":3100,"style":3042},[612],[574,3102,1591],{"className":3103},[617],[574,3105,3107,3110],{"className":3106},[617],[574,3108,714],{"className":3109},[617,618],[574,3111,3113],{"className":3112},[1185],[574,3114,3116,3136],{"className":3115},[822,1734],[574,3117,3119,3133],{"className":3118},[826],[574,3120,3122],{"className":3121,"style":1741},[830],[574,3123,3124,3127],{"style":1744},[574,3125],{"className":3126,"style":1201},[838],[574,3128,3130],{"className":3129},[1205,1206,1207,1208],[574,3131,798],{"className":3132},[617,1208],[574,3134,1758],{"className":3135},[1757],[574,3137,3139],{"className":3138},[826],[574,3140,3142],{"className":3141,"style":1765},[830],[574,3143],{},[574,3145],{"className":3146,"style":874},[737],[574,3148,721],{"className":3149},[878],[574,3151],{"className":3152,"style":874},[737],[574,3154,3156,3159,3162,3202,3205,3208],{"className":3155},[608],[574,3157],{"className":3158,"style":3042},[612],[574,3160,2814],{"className":3161},[617],[574,3163,3165,3168],{"className":3164},[617],[574,3166,714],{"className":3167},[617,618],[574,3169,3171],{"className":3170},[1185],[574,3172,3174,3194],{"className":3173},[822,1734],[574,3175,3177,3191],{"className":3176},[826],[574,3178,3180],{"className":3179,"style":1741},[830],[574,3181,3182,3185],{"style":1744},[574,3183],{"className":3184,"style":1201},[838],[574,3186,3188],{"className":3187},[1205,1206,1207,1208],[574,3189,1506],{"className":3190},[617,1208],[574,3192,1758],{"className":3193},[1757],[574,3195,3197],{"className":3196},[826],[574,3198,3200],{"className":3199,"style":1765},[830],[574,3201],{},[574,3203],{"className":3204,"style":874},[737],[574,3206,721],{"className":3207},[878],[574,3209],{"className":3210,"style":874},[737],[574,3212,3214,3217,3220,3223,3226],{"className":3213},[608],[574,3215],{"className":3216,"style":1622},[612],[574,3218,798],{"className":3219},[617],[574,3221],{"className":3222,"style":930},[737],[574,3224,733],{"className":3225},[934],[574,3227],{"className":3228,"style":930},[737],[574,3230,3232,3235],{"className":3231},[608],[574,3233],{"className":3234,"style":1622},[612],[574,3236,2153],{"className":3237},[617],". Quadratische Ergänzung:",[574,3240,3242],{"className":3241},[691],[574,3243,3245,3335],{"className":3244},[577],[574,3246,3248],{"className":3247},[581],[583,3249,3250],{"xmlns":585,"display":700},[587,3251,3252,3332],{},[590,3253,3254,3256,3262,3264,3266,3272,3274,3276,3278,3280,3286,3288,3290,3296,3298,3300,3302,3304,3310,3312,3314,3320,3322,3324,3326,3328,3330],{},[716,3255,749],{"stretchy":748},[1645,3257,3258,3260],{},[593,3259,714],{},[796,3261,1495],{},[716,3263,721],{},[796,3265,1495],{},[791,3267,3268,3270],{},[716,3269,766],{"stretchy":748},[796,3271,798],{},[716,3273,721],{},[796,3275,1495],{},[716,3277,1663],{},[716,3279,749],{"stretchy":748},[1645,3281,3282,3284],{},[593,3283,714],{},[796,3285,798],{},[716,3287,1663],{},[796,3289,798],{},[791,3291,3292,3294],{},[716,3293,766],{"stretchy":748},[796,3295,798],{},[716,3297,721],{},[796,3299,1591],{},[716,3301,1663],{},[716,3303,749],{"stretchy":748},[1645,3305,3306,3308],{},[593,3307,714],{},[796,3309,1506],{},[716,3311,721],{},[796,3313,1506],{},[791,3315,3316,3318],{},[716,3317,766],{"stretchy":748},[796,3319,798],{},[716,3321,721],{},[796,3323,2383],{},[716,3325,721],{},[796,3327,798],{},[716,3329,733],{},[796,3331,2153],{},[597,3333,3334],{"encoding":599},"(x_1 - 1)^2 - 1 + (x_2 + 2)^2 - 4 + (x_3 - 3)^2 - 9 - 2 = 0",[574,3336,3338,3396,3443,3461,3519,3566,3584,3642,3689,3707,3725],{"className":3337,"ariaHidden":604},[603],[574,3339,3341,3344,3347,3387,3390,3393],{"className":3340},[608],[574,3342],{"className":3343,"style":811},[612],[574,3345,749],{"className":3346},[964],[574,3348,3350,3353],{"className":3349},[617],[574,3351,714],{"className":3352},[617,618],[574,3354,3356],{"className":3355},[1185],[574,3357,3359,3379],{"className":3358},[822,1734],[574,3360,3362,3376],{"className":3361},[826],[574,3363,3365],{"className":3364,"style":1741},[830],[574,3366,3367,3370],{"style":1744},[574,3368],{"className":3369,"style":1201},[838],[574,3371,3373],{"className":3372},[1205,1206,1207,1208],[574,3374,1495],{"className":3375},[617,1208],[574,3377,1758],{"className":3378},[1757],[574,3380,3382],{"className":3381},[826],[574,3383,3385],{"className":3384,"style":1765},[830],[574,3386],{},[574,3388],{"className":3389,"style":874},[737],[574,3391,721],{"className":3392},[878],[574,3394],{"className":3395,"style":874},[737],[574,3397,3399,3402,3405,3434,3437,3440],{"className":3398},[608],[574,3400],{"className":3401,"style":1783},[612],[574,3403,1495],{"className":3404},[617],[574,3406,3408,3411],{"className":3407},[1053],[574,3409,766],{"className":3410},[1053],[574,3412,3414],{"className":3413},[1185],[574,3415,3417],{"className":3416},[822],[574,3418,3420],{"className":3419},[826],[574,3421,3423],{"className":3422,"style":1175},[830],[574,3424,3425,3428],{"style":1197},[574,3426],{"className":3427,"style":1201},[838],[574,3429,3431],{"className":3430},[1205,1206,1207,1208],[574,3432,798],{"className":3433},[617,1208],[574,3435],{"className":3436,"style":874},[737],[574,3438,721],{"className":3439},[878],[574,3441],{"className":3442,"style":874},[737],[574,3444,3446,3449,3452,3455,3458],{"className":3445},[608],[574,3447],{"className":3448,"style":2178},[612],[574,3450,1495],{"className":3451},[617],[574,3453],{"className":3454,"style":874},[737],[574,3456,1663],{"className":3457},[878],[574,3459],{"className":3460,"style":874},[737],[574,3462,3464,3467,3470,3510,3513,3516],{"className":3463},[608],[574,3465],{"className":3466,"style":811},[612],[574,3468,749],{"className":3469},[964],[574,3471,3473,3476],{"className":3472},[617],[574,3474,714],{"className":3475},[617,618],[574,3477,3479],{"className":3478},[1185],[574,3480,3482,3502],{"className":3481},[822,1734],[574,3483,3485,3499],{"className":3484},[826],[574,3486,3488],{"className":3487,"style":1741},[830],[574,3489,3490,3493],{"style":1744},[574,3491],{"className":3492,"style":1201},[838],[574,3494,3496],{"className":3495},[1205,1206,1207,1208],[574,3497,798],{"className":3498},[617,1208],[574,3500,1758],{"className":3501},[1757],[574,3503,3505],{"className":3504},[826],[574,3506,3508],{"className":3507,"style":1765},[830],[574,3509],{},[574,3511],{"className":3512,"style":874},[737],[574,3514,1663],{"className":3515},[878],[574,3517],{"className":3518,"style":874},[737],[574,3520,3522,3525,3528,3557,3560,3563],{"className":3521},[608],[574,3523],{"className":3524,"style":1783},[612],[574,3526,798],{"className":3527},[617],[574,3529,3531,3534],{"className":3530},[1053],[574,3532,766],{"className":3533},[1053],[574,3535,3537],{"className":3536},[1185],[574,3538,3540],{"className":3539},[822],[574,3541,3543],{"className":3542},[826],[574,3544,3546],{"className":3545,"style":1175},[830],[574,3547,3548,3551],{"style":1197},[574,3549],{"className":3550,"style":1201},[838],[574,3552,3554],{"className":3553},[1205,1206,1207,1208],[574,3555,798],{"className":3556},[617,1208],[574,3558],{"className":3559,"style":874},[737],[574,3561,721],{"className":3562},[878],[574,3564],{"className":3565,"style":874},[737],[574,3567,3569,3572,3575,3578,3581],{"className":3568},[608],[574,3570],{"className":3571,"style":2178},[612],[574,3573,1591],{"className":3574},[617],[574,3576],{"className":3577,"style":874},[737],[574,3579,1663],{"className":3580},[878],[574,3582],{"className":3583,"style":874},[737],[574,3585,3587,3590,3593,3633,3636,3639],{"className":3586},[608],[574,3588],{"className":3589,"style":811},[612],[574,3591,749],{"className":3592},[964],[574,3594,3596,3599],{"className":3595},[617],[574,3597,714],{"className":3598},[617,618],[574,3600,3602],{"className":3601},[1185],[574,3603,3605,3625],{"className":3604},[822,1734],[574,3606,3608,3622],{"className":3607},[826],[574,3609,3611],{"className":3610,"style":1741},[830],[574,3612,3613,3616],{"style":1744},[574,3614],{"className":3615,"style":1201},[838],[574,3617,3619],{"className":3618},[1205,1206,1207,1208],[574,3620,1506],{"className":3621},[617,1208],[574,3623,1758],{"className":3624},[1757],[574,3626,3628],{"className":3627},[826],[574,3629,3631],{"className":3630,"style":1765},[830],[574,3632],{},[574,3634],{"className":3635,"style":874},[737],[574,3637,721],{"className":3638},[878],[574,3640],{"className":3641,"style":874},[737],[574,3643,3645,3648,3651,3680,3683,3686],{"className":3644},[608],[574,3646],{"className":3647,"style":1783},[612],[574,3649,1506],{"className":3650},[617],[574,3652,3654,3657],{"className":3653},[1053],[574,3655,766],{"className":3656},[1053],[574,3658,3660],{"className":3659},[1185],[574,3661,3663],{"className":3662},[822],[574,3664,3666],{"className":3665},[826],[574,3667,3669],{"className":3668,"style":1175},[830],[574,3670,3671,3674],{"style":1197},[574,3672],{"className":3673,"style":1201},[838],[574,3675,3677],{"className":3676},[1205,1206,1207,1208],[574,3678,798],{"className":3679},[617,1208],[574,3681],{"className":3682,"style":874},[737],[574,3684,721],{"className":3685},[878],[574,3687],{"className":3688,"style":874},[737],[574,3690,3692,3695,3698,3701,3704],{"className":3691},[608],[574,3693],{"className":3694,"style":2178},[612],[574,3696,2383],{"className":3697},[617],[574,3699],{"className":3700,"style":874},[737],[574,3702,721],{"className":3703},[878],[574,3705],{"className":3706,"style":874},[737],[574,3708,3710,3713,3716,3719,3722],{"className":3709},[608],[574,3711],{"className":3712,"style":1622},[612],[574,3714,798],{"className":3715},[617],[574,3717],{"className":3718,"style":930},[737],[574,3720,733],{"className":3721},[934],[574,3723],{"className":3724,"style":930},[737],[574,3726,3728,3731],{"className":3727},[608],[574,3729],{"className":3730,"style":1622},[612],[574,3732,2153],{"className":3733},[617],[565,3735,3736,3737,4230,4231,1572,4323,1383],{},"Also ",[574,3738,3740,3830],{"className":3739},[577],[574,3741,3743],{"className":3742},[581],[583,3744,3745],{"xmlns":585},[587,3746,3747,3827],{},[590,3748,3749,3751,3757,3759,3761,3767,3769,3771,3777,3779,3781,3787,3789,3791,3797,3799,3801,3807,3809,3811,3813,3815,3817,3819,3821,3823,3825],{},[716,3750,749],{"stretchy":748},[1645,3752,3753,3755],{},[593,3754,714],{},[796,3756,1495],{},[716,3758,721],{},[796,3760,1495],{},[791,3762,3763,3765],{},[716,3764,766],{"stretchy":748},[796,3766,798],{},[716,3768,1663],{},[716,3770,749],{"stretchy":748},[1645,3772,3773,3775],{},[593,3774,714],{},[796,3776,798],{},[716,3778,1663],{},[796,3780,798],{},[791,3782,3783,3785],{},[716,3784,766],{"stretchy":748},[796,3786,798],{},[716,3788,1663],{},[716,3790,749],{"stretchy":748},[1645,3792,3793,3795],{},[593,3794,714],{},[796,3796,1506],{},[716,3798,721],{},[796,3800,1506],{},[791,3802,3803,3805],{},[716,3804,766],{"stretchy":748},[796,3806,798],{},[716,3808,733],{},[796,3810,798],{},[716,3812,1663],{},[796,3814,1495],{},[716,3816,1663],{},[796,3818,1591],{},[716,3820,1663],{},[796,3822,2383],{},[716,3824,733],{},[796,3826,1706],{},[597,3828,3829],{"encoding":599},"(x_1 - 1)^2 + (x_2 + 2)^2 + (x_3 - 3)^2 = 2 + 1 + 4 + 9 = 16",[574,3831,3833,3891,3939,3997,4044,4102,4149,4167,4185,4203,4221],{"className":3832,"ariaHidden":604},[603],[574,3834,3836,3839,3842,3882,3885,3888],{"className":3835},[608],[574,3837],{"className":3838,"style":811},[612],[574,3840,749],{"className":3841},[964],[574,3843,3845,3848],{"className":3844},[617],[574,3846,714],{"className":3847},[617,618],[574,3849,3851],{"className":3850},[1185],[574,3852,3854,3874],{"className":3853},[822,1734],[574,3855,3857,3871],{"className":3856},[826],[574,3858,3860],{"className":3859,"style":1741},[830],[574,3861,3862,3865],{"style":1744},[574,3863],{"className":3864,"style":1201},[838],[574,3866,3868],{"className":3867},[1205,1206,1207,1208],[574,3869,1495],{"className":3870},[617,1208],[574,3872,1758],{"className":3873},[1757],[574,3875,3877],{"className":3876},[826],[574,3878,3880],{"className":3879,"style":1765},[830],[574,3881],{},[574,3883],{"className":3884,"style":874},[737],[574,3886,721],{"className":3887},[878],[574,3889],{"className":3890,"style":874},[737],[574,3892,3894,3898,3901,3930,3933,3936],{"className":3893},[608],[574,3895],{"className":3896,"style":3897},[612],"height:1.0641em;vertical-align:-0.25em;",[574,3899,1495],{"className":3900},[617],[574,3902,3904,3907],{"className":3903},[1053],[574,3905,766],{"className":3906},[1053],[574,3908,3910],{"className":3909},[1185],[574,3911,3913],{"className":3912},[822],[574,3914,3916],{"className":3915},[826],[574,3917,3919],{"className":3918,"style":1267},[830],[574,3920,3921,3924],{"style":1288},[574,3922],{"className":3923,"style":1201},[838],[574,3925,3927],{"className":3926},[1205,1206,1207,1208],[574,3928,798],{"className":3929},[617,1208],[574,3931],{"className":3932,"style":874},[737],[574,3934,1663],{"className":3935},[878],[574,3937],{"className":3938,"style":874},[737],[574,3940,3942,3945,3948,3988,3991,3994],{"className":3941},[608],[574,3943],{"className":3944,"style":811},[612],[574,3946,749],{"className":3947},[964],[574,3949,3951,3954],{"className":3950},[617],[574,3952,714],{"className":3953},[617,618],[574,3955,3957],{"className":3956},[1185],[574,3958,3960,3980],{"className":3959},[822,1734],[574,3961,3963,3977],{"className":3962},[826],[574,3964,3966],{"className":3965,"style":1741},[830],[574,3967,3968,3971],{"style":1744},[574,3969],{"className":3970,"style":1201},[838],[574,3972,3974],{"className":3973},[1205,1206,1207,1208],[574,3975,798],{"className":3976},[617,1208],[574,3978,1758],{"className":3979},[1757],[574,3981,3983],{"className":3982},[826],[574,3984,3986],{"className":3985,"style":1765},[830],[574,3987],{},[574,3989],{"className":3990,"style":874},[737],[574,3992,1663],{"className":3993},[878],[574,3995],{"className":3996,"style":874},[737],[574,3998,4000,4003,4006,4035,4038,4041],{"className":3999},[608],[574,4001],{"className":4002,"style":3897},[612],[574,4004,798],{"className":4005},[617],[574,4007,4009,4012],{"className":4008},[1053],[574,4010,766],{"className":4011},[1053],[574,4013,4015],{"className":4014},[1185],[574,4016,4018],{"className":4017},[822],[574,4019,4021],{"className":4020},[826],[574,4022,4024],{"className":4023,"style":1267},[830],[574,4025,4026,4029],{"style":1288},[574,4027],{"className":4028,"style":1201},[838],[574,4030,4032],{"className":4031},[1205,1206,1207,1208],[574,4033,798],{"className":4034},[617,1208],[574,4036],{"className":4037,"style":874},[737],[574,4039,1663],{"className":4040},[878],[574,4042],{"className":4043,"style":874},[737],[574,4045,4047,4050,4053,4093,4096,4099],{"className":4046},[608],[574,4048],{"className":4049,"style":811},[612],[574,4051,749],{"className":4052},[964],[574,4054,4056,4059],{"className":4055},[617],[574,4057,714],{"className":4058},[617,618],[574,4060,4062],{"className":4061},[1185],[574,4063,4065,4085],{"className":4064},[822,1734],[574,4066,4068,4082],{"className":4067},[826],[574,4069,4071],{"className":4070,"style":1741},[830],[574,4072,4073,4076],{"style":1744},[574,4074],{"className":4075,"style":1201},[838],[574,4077,4079],{"className":4078},[1205,1206,1207,1208],[574,4080,1506],{"className":4081},[617,1208],[574,4083,1758],{"className":4084},[1757],[574,4086,4088],{"className":4087},[826],[574,4089,4091],{"className":4090,"style":1765},[830],[574,4092],{},[574,4094],{"className":4095,"style":874},[737],[574,4097,721],{"className":4098},[878],[574,4100],{"className":4101,"style":874},[737],[574,4103,4105,4108,4111,4140,4143,4146],{"className":4104},[608],[574,4106],{"className":4107,"style":3897},[612],[574,4109,1506],{"className":4110},[617],[574,4112,4114,4117],{"className":4113},[1053],[574,4115,766],{"className":4116},[1053],[574,4118,4120],{"className":4119},[1185],[574,4121,4123],{"className":4122},[822],[574,4124,4126],{"className":4125},[826],[574,4127,4129],{"className":4128,"style":1267},[830],[574,4130,4131,4134],{"style":1288},[574,4132],{"className":4133,"style":1201},[838],[574,4135,4137],{"className":4136},[1205,1206,1207,1208],[574,4138,798],{"className":4139},[617,1208],[574,4141],{"className":4142,"style":930},[737],[574,4144,733],{"className":4145},[934],[574,4147],{"className":4148,"style":930},[737],[574,4150,4152,4155,4158,4161,4164],{"className":4151},[608],[574,4153],{"className":4154,"style":2178},[612],[574,4156,798],{"className":4157},[617],[574,4159],{"className":4160,"style":874},[737],[574,4162,1663],{"className":4163},[878],[574,4165],{"className":4166,"style":874},[737],[574,4168,4170,4173,4176,4179,4182],{"className":4169},[608],[574,4171],{"className":4172,"style":2178},[612],[574,4174,1495],{"className":4175},[617],[574,4177],{"className":4178,"style":874},[737],[574,4180,1663],{"className":4181},[878],[574,4183],{"className":4184,"style":874},[737],[574,4186,4188,4191,4194,4197,4200],{"className":4187},[608],[574,4189],{"className":4190,"style":2178},[612],[574,4192,1591],{"className":4193},[617],[574,4195],{"className":4196,"style":874},[737],[574,4198,1663],{"className":4199},[878],[574,4201],{"className":4202,"style":874},[737],[574,4204,4206,4209,4212,4215,4218],{"className":4205},[608],[574,4207],{"className":4208,"style":1622},[612],[574,4210,2383],{"className":4211},[617],[574,4213],{"className":4214,"style":930},[737],[574,4216,733],{"className":4217},[934],[574,4219],{"className":4220,"style":930},[737],[574,4222,4224,4227],{"className":4223},[608],[574,4225],{"className":4226,"style":1622},[612],[574,4228,1706],{"className":4229},[617],", das ist dieselbe Kugel mit ",[574,4232,4234,4263],{"className":4233},[577],[574,4235,4237],{"className":4236},[581],[583,4238,4239],{"xmlns":585},[587,4240,4241,4261],{},[590,4242,4243,4245,4247,4249,4251,4253,4255,4257,4259],{},[593,4244,639],{},[716,4246,749],{"stretchy":748},[796,4248,1495],{},[716,4250,708],{},[716,4252,721],{},[796,4254,798],{},[716,4256,708],{},[796,4258,1506],{},[716,4260,766],{"stretchy":748},[597,4262,1511],{"encoding":599},[574,4264,4266,4290,4311],{"className":4265,"ariaHidden":604},[603],[574,4267,4269,4272,4275,4278,4281,4284,4287],{"className":4268},[608],[574,4270],{"className":4271,"style":811},[612],[574,4273,639],{"className":4274,"style":654},[617,618],[574,4276,749],{"className":4277},[964],[574,4279,1495],{"className":4280},[617],[574,4282],{"className":4283,"style":930},[737],[574,4285,708],{"className":4286},[934],[574,4288],{"className":4289,"style":930},[737],[574,4291,4293,4296,4299,4302,4305,4308],{"className":4292},[608],[574,4294],{"className":4295,"style":811},[612],[574,4297,721],{"className":4298},[617],[574,4300,798],{"className":4301},[617],[574,4303],{"className":4304,"style":930},[737],[574,4306,708],{"className":4307},[934],[574,4309],{"className":4310,"style":930},[737],[574,4312,4314,4317,4320],{"className":4313},[608],[574,4315],{"className":4316,"style":811},[612],[574,4318,1506],{"className":4319},[617],[574,4321,766],{"className":4322},[1053],[574,4324,4326,4343],{"className":4325},[577],[574,4327,4329],{"className":4328},[581],[583,4330,4331],{"xmlns":585},[587,4332,4333,4341],{},[590,4334,4335,4337,4339],{},[593,4336,670],{},[716,4338,733],{},[796,4340,1591],{},[597,4342,1594],{"encoding":599},[574,4344,4346,4364],{"className":4345,"ariaHidden":604},[603],[574,4347,4349,4352,4355,4358,4361],{"className":4348},[608],[574,4350],{"className":4351,"style":682},[612],[574,4353,670],{"className":4354,"style":686},[617,618],[574,4356],{"className":4357,"style":930},[737],[574,4359,733],{"className":4360},[934],[574,4362],{"className":4363,"style":930},[737],[574,4365,4367,4370],{"className":4366},[608],[574,4368],{"className":4369,"style":1622},[612],[574,4371,1591],{"className":4372},[617],[565,4374,4375],{},"Steht rechts nach der Ergänzung null, besteht die Lösungsmenge nur aus dem Mittelpunkt. Steht dort eine negative Zahl, gibt es keine Lösung, also keine Kugel.",[560,4377,4379],{"id":4378},"typische-fehler","Typische Fehler",[1216,4381,4382,4651,4821],{},[1219,4383,4384,4385,4517,4518,4616,4617,1383],{},"Vorzeichen des Mittelpunkts falsch ablesen: ",[574,4386,4388,4418],{"className":4387},[577],[574,4389,4391],{"className":4390},[581],[583,4392,4393],{"xmlns":585},[587,4394,4395,4415],{},[590,4396,4397,4399,4405,4407,4409],{},[716,4398,749],{"stretchy":748},[1645,4400,4401,4403],{},[593,4402,714],{},[796,4404,798],{},[716,4406,1663],{},[796,4408,798],{},[791,4410,4411,4413],{},[716,4412,766],{"stretchy":748},[796,4414,798],{},[597,4416,4417],{"encoding":599},"(x_2 + 2)^2",[574,4419,4421,4479],{"className":4420,"ariaHidden":604},[603],[574,4422,4424,4427,4430,4470,4473,4476],{"className":4423},[608],[574,4425],{"className":4426,"style":811},[612],[574,4428,749],{"className":4429},[964],[574,4431,4433,4436],{"className":4432},[617],[574,4434,714],{"className":4435},[617,618],[574,4437,4439],{"className":4438},[1185],[574,4440,4442,4462],{"className":4441},[822,1734],[574,4443,4445,4459],{"className":4444},[826],[574,4446,4448],{"className":4447,"style":1741},[830],[574,4449,4450,4453],{"style":1744},[574,4451],{"className":4452,"style":1201},[838],[574,4454,4456],{"className":4455},[1205,1206,1207,1208],[574,4457,798],{"className":4458},[617,1208],[574,4460,1758],{"className":4461},[1757],[574,4463,4465],{"className":4464},[826],[574,4466,4468],{"className":4467,"style":1765},[830],[574,4469],{},[574,4471],{"className":4472,"style":874},[737],[574,4474,1663],{"className":4475},[878],[574,4477],{"className":4478,"style":874},[737],[574,4480,4482,4485,4488],{"className":4481},[608],[574,4483],{"className":4484,"style":3897},[612],[574,4486,798],{"className":4487},[617],[574,4489,4491,4494],{"className":4490},[1053],[574,4492,766],{"className":4493},[1053],[574,4495,4497],{"className":4496},[1185],[574,4498,4500],{"className":4499},[822],[574,4501,4503],{"className":4502},[826],[574,4504,4506],{"className":4505,"style":1267},[830],[574,4507,4508,4511],{"style":1288},[574,4509],{"className":4510,"style":1201},[838],[574,4512,4514],{"className":4513},[1205,1206,1207,1208],[574,4515,798],{"className":4516},[617,1208]," gehört zu ",[574,4519,4521,4545],{"className":4520},[577],[574,4522,4524],{"className":4523},[581],[583,4525,4526],{"xmlns":585},[587,4527,4528,4542],{},[590,4529,4530,4536,4538,4540],{},[1645,4531,4532,4534],{},[593,4533,726],{},[796,4535,798],{},[716,4537,733],{},[716,4539,721],{},[796,4541,798],{},[597,4543,4544],{"encoding":599},"m_2 = -2",[574,4546,4548,4604],{"className":4547,"ariaHidden":604},[603],[574,4549,4551,4555,4595,4598,4601],{"className":4550},[608],[574,4552],{"className":4553,"style":4554},[612],"height:0.5806em;vertical-align:-0.15em;",[574,4556,4558,4561],{"className":4557},[617],[574,4559,726],{"className":4560},[617,618],[574,4562,4564],{"className":4563},[1185],[574,4565,4567,4587],{"className":4566},[822,1734],[574,4568,4570,4584],{"className":4569},[826],[574,4571,4573],{"className":4572,"style":1741},[830],[574,4574,4575,4578],{"style":1744},[574,4576],{"className":4577,"style":1201},[838],[574,4579,4581],{"className":4580},[1205,1206,1207,1208],[574,4582,798],{"className":4583},[617,1208],[574,4585,1758],{"className":4586},[1757],[574,4588,4590],{"className":4589},[826],[574,4591,4593],{"className":4592,"style":1765},[830],[574,4594],{},[574,4596],{"className":4597,"style":930},[737],[574,4599,733],{"className":4600},[934],[574,4602],{"className":4603,"style":930},[737],[574,4605,4607,4610,4613],{"className":4606},[608],[574,4608],{"className":4609,"style":2178},[612],[574,4611,721],{"className":4612},[617],[574,4614,798],{"className":4615},[617],", nicht ",[574,4618,4620,4636],{"className":4619},[577],[574,4621,4623],{"className":4622},[581],[583,4624,4625],{"xmlns":585},[587,4626,4627,4633],{},[590,4628,4629,4631],{},[716,4630,1663],{},[796,4632,798],{},[597,4634,4635],{"encoding":599},"+2",[574,4637,4639],{"className":4638,"ariaHidden":604},[603],[574,4640,4642,4645,4648],{"className":4641},[608],[574,4643],{"className":4644,"style":2178},[612],[574,4646,1663],{"className":4647},[617],[574,4649,798],{"className":4650},[617],[1219,4652,4653,4681,4682,4741,4742,4792,4793,1383],{},[574,4654,4656,4669],{"className":4655},[577],[574,4657,4659],{"className":4658},[581],[583,4660,4661],{"xmlns":585},[587,4662,4663,4667],{},[590,4664,4665],{},[593,4666,670],{},[597,4668,670],{"encoding":599},[574,4670,4672],{"className":4671,"ariaHidden":604},[603],[574,4673,4675,4678],{"className":4674},[608],[574,4676],{"className":4677,"style":682},[612],[574,4679,670],{"className":4680,"style":686},[617,618]," statt ",[574,4683,4685,4703],{"className":4684},[577],[574,4686,4688],{"className":4687},[581],[583,4689,4690],{"xmlns":585},[587,4691,4692,4700],{},[590,4693,4694],{},[791,4695,4696,4698],{},[593,4697,670],{},[796,4699,798],{},[597,4701,4702],{"encoding":599},"r^2",[574,4704,4706],{"className":4705,"ariaHidden":604},[603],[574,4707,4709,4712],{"className":4708},[608],[574,4710],{"className":4711,"style":1267},[612],[574,4713,4715,4718],{"className":4714},[617],[574,4716,670],{"className":4717,"style":686},[617,618],[574,4719,4721],{"className":4720},[1185],[574,4722,4724],{"className":4723},[822],[574,4725,4727],{"className":4726},[826],[574,4728,4730],{"className":4729,"style":1267},[830],[574,4731,4732,4735],{"style":1288},[574,4733],{"className":4734,"style":1201},[838],[574,4736,4738],{"className":4737},[1205,1206,1207,1208],[574,4739,798],{"className":4740},[617,1208]," auf die rechte Seite schreiben. Bei ",[574,4743,4745,4762],{"className":4744},[577],[574,4746,4748],{"className":4747},[581],[583,4749,4750],{"xmlns":585},[587,4751,4752,4760],{},[590,4753,4754,4756,4758],{},[593,4755,670],{},[716,4757,733],{},[796,4759,1591],{},[597,4761,1594],{"encoding":599},[574,4763,4765,4783],{"className":4764,"ariaHidden":604},[603],[574,4766,4768,4771,4774,4777,4780],{"className":4767},[608],[574,4769],{"className":4770,"style":682},[612],[574,4772,670],{"className":4773,"style":686},[617,618],[574,4775],{"className":4776,"style":930},[737],[574,4778,733],{"className":4779},[934],[574,4781],{"className":4782,"style":930},[737],[574,4784,4786,4789],{"className":4785},[608],[574,4787],{"className":4788,"style":1622},[612],[574,4790,1591],{"className":4791},[617]," steht dort ",[574,4794,4796,4809],{"className":4795},[577],[574,4797,4799],{"className":4798},[581],[583,4800,4801],{"xmlns":585},[587,4802,4803,4807],{},[590,4804,4805],{},[796,4806,1706],{},[597,4808,1706],{"encoding":599},[574,4810,4812],{"className":4811,"ariaHidden":604},[603],[574,4813,4815,4818],{"className":4814},[608],[574,4816],{"className":4817,"style":1622},[612],[574,4819,1706],{"className":4820},[617],[1219,4822,4823],{},"Bei der quadratischen Ergänzung die abgezogenen Quadrate vergessen, sodass der Radius zu klein herauskommt.",{"title":4825,"searchDepth":4826,"depth":4826,"links":4827},"",2,[4828,4829,4830],{"id":562,"depth":4826,"text":563},{"id":1472,"depth":4826,"text":1473},{"id":4378,"depth":4826,"text":4379},[4832,4835,4837,4839,4841,4844],{"lhs":4833,"rhs":4834},"(x1 - m1)^2 + (x2 - m2)^2 + (x3 - m3)^2 - r^2","x1^2 + x2^2 + x3^2 - 2*m1*x1 - 2*m2*x2 - 2*m3*x3 + m1^2 + m2^2 + m3^2 - r^2",{"lhs":4836,"rhs":1706},"(1 - 1)^2 + (2 + 2)^2 + (3 - 3)^2",{"lhs":4838,"rhs":2383},"(3 - 1)^2 + (0 + 2)^2 + (4 - 3)^2",{"lhs":4840,"rhs":2625},"(5 - 1)^2 + (2 + 2)^2 + (3 - 3)^2",{"lhs":4842,"rhs":4843},"(x1 - 1)^2 + (x2 + 2)^2 + (x3 - 3)^2 - 16","x1^2 + x2^2 + x3^2 - 2*x1 + 4*x2 - 6*x3 - 2",{"lhs":4845,"rhs":1706},"2 + 1 + 4 + 9","r > 0","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>r\u003C\u002Fmi>\u003Cmo>>\u003C\u002Fmo>\u003Cmn>0\u003C\u002Fmn>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">r > 0\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.5782em;vertical-align:-0.0391em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0278em;\">r\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.6444em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">0\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",null,"md","sphere-equation","(x_1 - m_1)^2 + (x_2 - m_2)^2 + (x_3 - m_3)^2 = r^2","\u003Cspan class=\"katex-display\">\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\" display=\"block\">\u003Csemantics>\u003Cmrow>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmsub>\u003Cmi>x\u003C\u002Fmi>\u003Cmn>1\u003C\u002Fmn>\u003C\u002Fmsub>\u003Cmo>−\u003C\u002Fmo>\u003Cmsub>\u003Cmi>m\u003C\u002Fmi>\u003Cmn>1\u003C\u002Fmn>\u003C\u002Fmsub>\u003Cmsup>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmn>2\u003C\u002Fmn>\u003C\u002Fmsup>\u003Cmo>+\u003C\u002Fmo>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmsub>\u003Cmi>x\u003C\u002Fmi>\u003Cmn>2\u003C\u002Fmn>\u003C\u002Fmsub>\u003Cmo>−\u003C\u002Fmo>\u003Cmsub>\u003Cmi>m\u003C\u002Fmi>\u003Cmn>2\u003C\u002Fmn>\u003C\u002Fmsub>\u003Cmsup>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmn>2\u003C\u002Fmn>\u003C\u002Fmsup>\u003Cmo>+\u003C\u002Fmo>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmsub>\u003Cmi>x\u003C\u002Fmi>\u003Cmn>3\u003C\u002Fmn>\u003C\u002Fmsub>\u003Cmo>−\u003C\u002Fmo>\u003Cmsub>\u003Cmi>m\u003C\u002Fmi>\u003Cmn>3\u003C\u002Fmn>\u003C\u002Fmsub>\u003Cmsup>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003Cmn>2\u003C\u002Fmn>\u003C\u002Fmsup>\u003Cmo>=\u003C\u002Fmo>\u003Cmsup>\u003Cmi>r\u003C\u002Fmi>\u003Cmn>2\u003C\u002Fmn>\u003C\u002Fmsup>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">(x_1 - m_1)^2 + (x_2 - m_2)^2 + (x_3 - m_3)^2 = r^2\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">x\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3011em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">1\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">−\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:1.1141em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">m\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3011em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">1\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose\">\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8641em;\">\u003Cspan style=\"top:-3.113em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">2\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">+\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">x\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3011em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">2\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">−\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:1.1141em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">m\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3011em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">2\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose\">\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8641em;\">\u003Cspan style=\"top:-3.113em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">2\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">+\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">x\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3011em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">3\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">−\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:1.1141em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">m\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3011em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">3\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose\">\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8641em;\">\u003Cspan style=\"top:-3.113em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">2\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.8641em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0278em;\">r\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.8641em;\">\u003Cspan style=\"top:-3.113em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">2\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>","upper",{},true,"\u002Fde\u002Fformulas\u002Fmath\u002Flinear-algebra\u002Fkugelgleichung",[],[4859,4860,4861,4862],"circle-equation","distance-two-points","vector-length","completing-the-square",{"title":548,"description":4825},"de\u002Fformulas\u002Fmath\u002Flinear-algebra\u002Fkugelgleichung","Eine Kugel besteht aus allen Punkten, die vom Mittelpunkt M genau den Abstand r haben.",[4867,4868,4869,4870,4871],"kugel","kugelgleichung","mittelpunkt","radius","analytische geometrie",[4873,4877,4881],{"symbol":4874,"meaning":4875,"symbolHtml":4876},"x_1, x_2, x_3","Koordinaten eines Punkts auf der Kugeloberfläche","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsub>\u003Cmi>x\u003C\u002Fmi>\u003Cmn>1\u003C\u002Fmn>\u003C\u002Fmsub>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmsub>\u003Cmi>x\u003C\u002Fmi>\u003Cmn>2\u003C\u002Fmn>\u003C\u002Fmsub>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmsub>\u003Cmi>x\u003C\u002Fmi>\u003Cmn>3\u003C\u002Fmn>\u003C\u002Fmsub>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">x_1, x_2, x_3\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.625em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">x\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3011em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">1\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">x\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3011em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">2\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mpunct\">,\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">x\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3011em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">3\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",{"symbol":4878,"meaning":4879,"symbolHtml":4880},"M(m_1 \\mid m_2 \\mid m_3)","Mittelpunkt der Kugel","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>M\u003C\u002Fmi>\u003Cmo stretchy=\"false\">(\u003C\u002Fmo>\u003Cmsub>\u003Cmi>m\u003C\u002Fmi>\u003Cmn>1\u003C\u002Fmn>\u003C\u002Fmsub>\u003Cmo>∣\u003C\u002Fmo>\u003Cmsub>\u003Cmi>m\u003C\u002Fmi>\u003Cmn>2\u003C\u002Fmn>\u003C\u002Fmsub>\u003Cmo>∣\u003C\u002Fmo>\u003Cmsub>\u003Cmi>m\u003C\u002Fmi>\u003Cmn>3\u003C\u002Fmn>\u003C\u002Fmsub>\u003Cmo stretchy=\"false\">)\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">M(m_1 \\mid m_2 \\mid m_3)\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.109em;\">M\u003C\u002Fspan>\u003Cspan class=\"mopen\">(\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">m\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3011em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">1\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">∣\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">m\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3011em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">2\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">∣\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:1em;vertical-align:-0.25em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord mathnormal\">m\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.3011em;\">\u003Cspan style=\"top:-2.55em;margin-left:0em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mord mtight\">3\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.15em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",{"symbol":670,"meaning":4882,"symbolHtml":4883},"Radius der Kugel","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>r\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">r\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.4306em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0278em;\">r\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>","zAezrIasiq7_uKXgjfe9ugL05HmnP07zvH12TKzKerY",[4886,4889,4892,4895],{"stem":4887,"title":4888,"category":5},"de\u002Fformulas\u002Fmath\u002Falgebra\u002Fquadratische-ergaenzung","Quadratische Ergänzung",{"stem":4890,"title":4891,"category":208},"de\u002Fformulas\u002Fmath\u002Ffunctions\u002Fabstand-zweier-punkte","Abstand zweier Punkte",{"stem":4893,"title":4894,"category":247},"de\u002Fformulas\u002Fmath\u002Fgeometry\u002Fkreisgleichung","Kreisgleichung",{"stem":4896,"title":4897,"category":312},"de\u002Fformulas\u002Fmath\u002Flinear-algebra\u002Fbetrag-eines-vektors","Betrag eines Vektors",1793824278583]