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0",[571,1161,1163,1181],{"className":1162,"ariaHidden":633},[632],[571,1164,1166,1169,1172,1175,1178],{"className":1165},[637],[571,1167],{"className":1168,"style":907},[641],[571,1170,868],{"className":1171,"style":911},[646,650],[571,1173],{"className":1174,"style":734},[689],[571,1176,620],{"className":1177},[738],[571,1179],{"className":1180,"style":734},[689],[571,1182,1184,1187],{"className":1183},[637],[571,1185],{"className":1186,"style":748},[641],[571,1188,623],{"className":1189},[646],": eine Lösung",[1081,1192,1193,1245],{},[571,1194,1196,1215],{"className":1195},[578],[571,1197,1199],{"className":1198},[582],[584,1200,1201],{"xmlns":586},[589,1202,1203,1212],{},[592,1204,1205,1207,1210],{},[598,1206,868],{},[606,1208,1209],{},"\u003C",[602,1211,623],{},[625,1213,1214],{"encoding":627},"D \u003C 0",[571,1216,1218,1236],{"className":1217,"ariaHidden":633},[632],[571,1219,1221,1224,1227,1230,1233],{"className":1220},[637],[571,1222],{"className":1223,"style":1114},[641],[571,1225,868],{"className":1226,"style":911},[646,650],[571,1228],{"className":1229,"style":734},[689],[571,1231,1209],{"className":1232},[738],[571,1234],{"className":1235,"style":734},[689],[571,1237,1239,1242],{"className":1238},[637],[571,1240],{"className":1241,"style":748},[641],[571,1243,623],{"className":1244},[646],": keine reelle Lösung",[557,1247,1249],{"id":1248},"herleitung","Herleitung",[562,1251,1252],{},"Die Formel entsteht durch quadratische Ergänzung mit der ersten binomischen Formel:",[571,1254,1256],{"className":1255},[574],[571,1257,1259,1342],{"className":1258},[578],[571,1260,1262],{"className":1261},[582],[584,1263,1264],{"xmlns":586,"display":587},[589,1265,1266,1339],{},[592,1267,1268,1274,1276,1278,1280,1282,1284,1286,1288,1292,1295,1297,1317,1319,1335,1337],{},[595,1269,1270,1272],{},[598,1271,600],{},[602,1273,604],{},[606,1275,608],{},[598,1277,562],{},[598,1279,600],{},[606,1281,608],{},[598,1283,617],{},[606,1285,620],{},[602,1287,623],{},[1289,1290,1291],"mtext",{},"  ",[606,1293,1294],{},"⇔",[1289,1296,1291],{},[595,1298,1299,1315],{},[592,1300,1301,1303,1305,1307,1313],{},[606,1302,877],{"fence":633},[598,1304,600],{},[606,1306,608],{},[879,1308,1309,1311],{},[598,1310,562],{},[602,1312,604],{},[606,1314,887],{"fence":633},[602,1316,604],{},[606,1318,620],{},[595,1320,1321,1333],{},[592,1322,1323,1325,1331],{},[606,1324,877],{"fence":633},[879,1326,1327,1329],{},[598,1328,562],{},[602,1330,604],{},[606,1332,887],{"fence":633},[602,1334,604],{},[606,1336,892],{},[598,1338,617],{},[625,1340,1341],{"encoding":627},"x^2 + px + q = 0\n\\;\\Leftrightarrow\\;\n\\left(x + \\frac{p}{2}\\right)^2 = \\left(\\frac{p}{2}\\right)^2 - q",[571,1343,1345,1389,1410,1428,1452,1590,1708],{"className":1344,"ariaHidden":633},[632],[571,1346,1348,1351,1380,1383,1386],{"className":1347},[637],[571,1349],{"className":1350,"style":642},[641],[571,1352,1354,1357],{"className":1353},[646],[571,1355,600],{"className":1356},[646,650],[571,1358,1360],{"className":1359},[654],[571,1361,1363],{"className":1362},[658],[571,1364,1366],{"className":1365},[662],[571,1367,1369],{"className":1368,"style":667},[666],[571,1370,1371,1374],{"style":670},[571,1372],{"className":1373,"style":675},[674],[571,1375,1377],{"className":1376},[679,680,681,682],[571,1378,604],{"className":1379},[646,682],[571,1381],{"className":1382,"style":690},[689],[571,1384,608],{"className":1385},[694],[571,1387],{"className":1388,"style":690},[689],[571,1390,1392,1395,1398,1401,1404,1407],{"className":1391},[637],[571,1393],{"className":1394,"style":704},[641],[571,1396,562],{"className":1397},[646,650],[571,1399,600],{"className":1400},[646,650],[571,1402],{"className":1403,"style":690},[689],[571,1405,608],{"className":1406},[694],[571,1408],{"className":1409,"style":690},[689],[571,1411,1413,1416,1419,1422,1425],{"className":1412},[637],[571,1414],{"className":1415,"style":726},[641],[571,1417,617],{"className":1418,"style":730},[646,650],[571,1420],{"className":1421,"style":734},[689],[571,1423,620],{"className":1424},[738],[571,1426],{"className":1427,"style":734},[689],[571,1429,1431,1434,1437,1440,1443,1446,1449],{"className":1430},[637],[571,1432],{"className":1433,"style":748},[641],[571,1435,623],{"className":1436},[646],[571,1438],{"className":1439,"style":734},[689],[571,1441],{"className":1442,"style":734},[689],[571,1444,1294],{"className":1445},[738],[571,1447],{"className":1448,"style":734},[689],[571,1450],{"className":1451,"style":734},[689],[571,1453,1455,1459,1581,1584,1587],{"className":1454},[637],[571,1456],{"className":1457,"style":1458},[641],"height:2.04em;vertical-align:-0.686em;",[571,1460,1462,1556],{"className":1461},[931],[571,1463,1465,1472,1475,1478,1481,1484,1550],{"className":1464},[931],[571,1466,1468],{"className":1467,"style":940},[938,939],[571,1469,877],{"className":1470},[944,1471],"size2",[571,1473,600],{"className":1474},[646,650],[571,1476],{"className":1477,"style":690},[689],[571,1479,608],{"className":1480},[694],[571,1482],{"className":1483,"style":690},[689],[571,1485,1487,1490,1547],{"className":1486},[646],[571,1488],{"className":1489},[938,952],[571,1491,1493],{"className":1492},[879],[571,1494,1496,1538],{"className":1495},[658,959],[571,1497,1499,1535],{"className":1498},[662],[571,1500,1503,1515,1523],{"className":1501,"style":1502},[666],"height:1.1076em;",[571,1504,1506,1509],{"style":1505},"top:-2.314em;",[571,1507],{"className":1508,"style":973},[674],[571,1510,1512],{"className":1511},[646],[571,1513,604],{"className":1514},[646],[571,1516,1517,1520],{"style":985},[571,1518],{"className":1519,"style":973},[674],[571,1521],{"className":1522,"style":993},[992],[571,1524,1526,1529],{"style":1525},"top:-3.677em;",[571,1527],{"className":1528,"style":973},[674],[571,1530,1532],{"className":1531},[646],[571,1533,562],{"className":1534},[646,650],[571,1536,1013],{"className":1537},[1012],[571,1539,1541],{"className":1540},[662],[571,1542,1545],{"className":1543,"style":1544},[666],"height:0.686em;",[571,1546],{},[571,1548],{"className":1549},[1026,952],[571,1551,1553],{"className":1552,"style":940},[1026,939],[571,1554,887],{"className":1555},[944,1471],[571,1557,1559],{"className":1558},[654],[571,1560,1562],{"className":1561},[658],[571,1563,1565],{"className":1564},[662],[571,1566,1569],{"className":1567,"style":1568},[666],"height:1.354em;",[571,1570,1572,1575],{"style":1571},"top:-3.6029em;margin-right:0.05em;",[571,1573],{"className":1574,"style":675},[674],[571,1576,1578],{"className":1577},[679,680,681,682],[571,1579,604],{"className":1580},[646,682],[571,1582],{"className":1583,"style":734},[689],[571,1585,620],{"className":1586},[738],[571,1588],{"className":1589,"style":734},[689],[571,1591,1593,1596,1699,1702,1705],{"className":1592},[637],[571,1594],{"className":1595,"style":1458},[641],[571,1597,1599,1676],{"className":1598},[931],[571,1600,1602,1608,1670],{"className":1601},[931],[571,1603,1605],{"className":1604,"style":940},[938,939],[571,1606,877],{"className":1607},[944,1471],[571,1609,1611,1614,1667],{"className":1610},[646],[571,1612],{"className":1613},[938,952],[571,1615,1617],{"className":1616},[879],[571,1618,1620,1659],{"className":1619},[658,959],[571,1621,1623,1656],{"className":1622},[662],[571,1624,1626,1637,1645],{"className":1625,"style":1502},[666],[571,1627,1628,1631],{"style":1505},[571,1629],{"className":1630,"style":973},[674],[571,1632,1634],{"className":1633},[646],[571,1635,604],{"className":1636},[646],[571,1638,1639,1642],{"style":985},[571,1640],{"className":1641,"style":973},[674],[571,1643],{"className":1644,"style":993},[992],[571,1646,1647,1650],{"style":1525},[571,1648],{"className":1649,"style":973},[674],[571,1651,1653],{"className":1652},[646],[571,1654,562],{"className":1655},[646,650],[571,1657,1013],{"className":1658},[1012],[571,1660,1662],{"className":1661},[662],[571,1663,1665],{"className":1664,"style":1544},[666],[571,1666],{},[571,1668],{"className":1669},[1026,952],[571,1671,1673],{"className":1672,"style":940},[1026,939],[571,1674,887],{"className":1675},[944,1471],[571,1677,1679],{"className":1678},[654],[571,1680,1682],{"className":1681},[658],[571,1683,1685],{"className":1684},[662],[571,1686,1688],{"className":1687,"style":1568},[666],[571,1689,1690,1693],{"style":1571},[571,1691],{"className":1692,"style":675},[674],[571,1694,1696],{"className":1695},[679,680,681,682],[571,1697,604],{"className":1698},[646,682],[571,1700],{"className":1701,"style":690},[689],[571,1703,892],{"className":1704},[694],[571,1706],{"className":1707,"style":690},[689],[571,1709,1711,1714],{"className":1710},[637],[571,1712],{"className":1713,"style":726},[641],[571,1715,617],{"className":1716,"style":730},[646,650],[562,1718,1719,1720,1749],{},"Wurzelziehen und Auflösen nach ",[571,1721,1723,1736],{"className":1722},[578],[571,1724,1726],{"className":1725},[582],[584,1727,1728],{"xmlns":586},[589,1729,1730,1734],{},[592,1731,1732],{},[598,1733,600],{},[625,1735,600],{"encoding":627},[571,1737,1739],{"className":1738,"ariaHidden":633},[632],[571,1740,1742,1746],{"className":1741},[637],[571,1743],{"className":1744,"style":1745},[641],"height:0.4306em;",[571,1747,600],{"className":1748},[646,650]," ergibt die p-q-Formel.",[557,1751,1753],{"id":1752},"beispiele","Beispiele",[562,1755,1756,1757,1891,1892,1948,1949,1076],{},"Gesucht sind die Lösungen von ",[571,1758,1760,1794],{"className":1759},[578],[571,1761,1763],{"className":1762},[582],[584,1764,1765],{"xmlns":586},[589,1766,1767,1791],{},[592,1768,1769,1775,1777,1780,1782,1784,1787,1789],{},[595,1770,1771,1773],{},[598,1772,600],{},[602,1774,604],{},[606,1776,892],{},[602,1778,1779],{},"5",[598,1781,600],{},[606,1783,608],{},[602,1785,1786],{},"6",[606,1788,620],{},[602,1790,623],{},[625,1792,1793],{"encoding":627},"x^2 - 5x + 6 = 0",[571,1795,1797,1842,1864,1882],{"className":1796,"ariaHidden":633},[632],[571,1798,1800,1804,1833,1836,1839],{"className":1799},[637],[571,1801],{"className":1802,"style":1803},[641],"height:0.8974em;vertical-align:-0.0833em;",[571,1805,1807,1810],{"className":1806},[646],[571,1808,600],{"className":1809},[646,650],[571,1811,1813],{"className":1812},[654],[571,1814,1816],{"className":1815},[658],[571,1817,1819],{"className":1818},[662],[571,1820,1822],{"className":1821,"style":785},[666],[571,1823,1824,1827],{"style":806},[571,1825],{"className":1826,"style":675},[674],[571,1828,1830],{"className":1829},[679,680,681,682],[571,1831,604],{"className":1832},[646,682],[571,1834],{"className":1835,"style":690},[689],[571,1837,892],{"className":1838},[694],[571,1840],{"className":1841,"style":690},[689],[571,1843,1845,1849,1852,1855,1858,1861],{"className":1844},[637],[571,1846],{"className":1847,"style":1848},[641],"height:0.7278em;vertical-align:-0.0833em;",[571,1850,1779],{"className":1851},[646],[571,1853,600],{"className":1854},[646,650],[571,1856],{"className":1857,"style":690},[689],[571,1859,608],{"className":1860},[694],[571,1862],{"className":1863,"style":690},[689],[571,1865,1867,1870,1873,1876,1879],{"className":1866},[637],[571,1868],{"className":1869,"style":748},[641],[571,1871,1786],{"className":1872},[646],[571,1874],{"className":1875,"style":734},[689],[571,1877,620],{"className":1878},[738],[571,1880],{"className":1881,"style":734},[689],[571,1883,1885,1888],{"className":1884},[637],[571,1886],{"className":1887,"style":748},[641],[571,1889,623],{"className":1890},[646],", also ",[571,1893,1895,1915],{"className":1894},[578],[571,1896,1898],{"className":1897},[582],[584,1899,1900],{"xmlns":586},[589,1901,1902,1912],{},[592,1903,1904,1906,1908,1910],{},[598,1905,562],{},[606,1907,620],{},[606,1909,892],{},[602,1911,1779],{},[625,1913,1914],{"encoding":627},"p = -5",[571,1916,1918,1936],{"className":1917,"ariaHidden":633},[632],[571,1919,1921,1924,1927,1930,1933],{"className":1920},[637],[571,1922],{"className":1923,"style":726},[641],[571,1925,562],{"className":1926},[646,650],[571,1928],{"className":1929,"style":734},[689],[571,1931,620],{"className":1932},[738],[571,1934],{"className":1935,"style":734},[689],[571,1937,1939,1942,1945],{"className":1938},[637],[571,1940],{"className":1941,"style":1848},[641],[571,1943,892],{"className":1944},[646],[571,1946,1779],{"className":1947},[646]," und ",[571,1950,1952,1970],{"className":1951},[578],[571,1953,1955],{"className":1954},[582],[584,1956,1957],{"xmlns":586},[589,1958,1959,1967],{},[592,1960,1961,1963,1965],{},[598,1962,617],{},[606,1964,620],{},[602,1966,1786],{},[625,1968,1969],{"encoding":627},"q = 6",[571,1971,1973,1991],{"className":1972,"ariaHidden":633},[632],[571,1974,1976,1979,1982,1985,1988],{"className":1975},[637],[571,1977],{"className":1978,"style":726},[641],[571,1980,617],{"className":1981,"style":730},[646,650],[571,1983],{"className":1984,"style":734},[689],[571,1986,620],{"className":1987},[738],[571,1989],{"className":1990,"style":734},[689],[571,1992,1994,1997],{"className":1993},[637],[571,1995],{"className":1996,"style":748},[641],[571,1998,1786],{"className":1999},[646],[571,2001,2003],{"className":2002},[574],[571,2004,2006,2070],{"className":2005},[578],[571,2007,2009],{"className":2008},[582],[584,2010,2011],{"xmlns":586,"display":587},[589,2012,2013,2067],{},[592,2014,2015,2029,2031,2037,2040,2057,2059,2062,2064],{},[2016,2017,2018,2020],"msub",{},[598,2019,600],{},[592,2021,2022,2024,2027],{},[602,2023,831],{},[606,2025,2026],{"separator":633},",",[602,2028,604],{},[606,2030,620],{},[879,2032,2033,2035],{},[602,2034,1779],{},[602,2036,604],{},[606,2038,2039],{},"±",[2041,2042,2043],"msqrt",{},[592,2044,2045,2053,2055],{},[879,2046,2047,2050],{},[602,2048,2049],{},"25",[602,2051,2052],{},"4",[606,2054,892],{},[602,2056,1786],{},[606,2058,620],{},[602,2060,2061],{},"2,5",[606,2063,2039],{},[602,2065,2066],{},"0,5",[625,2068,2069],{"encoding":627},"x_{1,2} = \\frac{5}{2} \\pm \\sqrt{\\frac{25}{4} - 6} = 2{,}5 \\pm 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