125:@0.938423:0.974518:0.967620:0.974518:0.967620:0.956590:0.938423:0.956590:0.000000:0.000000:0.000000 Pensamiento numérico y variacional :@0.603816:0.043660:0.930949:0.043660:0.930949:0.024915:0.603816:0.024915:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Análisis del discriminante :@0.073089:0.136836:0.269439:0.136836:0.269439:0.120167:0.073089:0.120167:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 La ecuación de segundo grado :@0.073089:0.164198:0.302417:0.164198:0.302417:0.147764:0.073089:0.147764:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 ax :@0.302149:0.164198:0.322029:0.164198:0.322029:0.147792:0.302149:0.147792:0.000000:0.000000:0.000000 ² + + = 0 tiene dos soluciones: :@0.321973:0.164198:0.576012:0.164198:0.576012:0.147764:0.321973:0.147764:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 bx:@0.345940:0.164198:0.362414:0.164198:0.362414:0.147792:0.345940:0.147792:0.000000:0.000000 c:@0.381153:0.164198:0.388810:0.164198:0.388810:0.147792:0.381153:0.147792:0.000000 La cantidad subradical ² – 4 se denomina discriminante, porque sirve para discriminar :@0.073089:0.246662:0.710015:0.246662:0.710015:0.230228:0.073089:0.230228:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 b :@0.236065:0.246662:0.248509:0.246662:0.248509:0.230256:0.236065:0.230256:0.000000:0.000000 ac:@0.277905:0.246662:0.294693:0.246662:0.294693:0.230256:0.277905:0.230256:0.000000:0.000000 (discernir) entre los tipos de soluciones.:@0.073088:0.263304:0.357772:0.263304:0.357772:0.246870:0.073088:0.246870:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Existen tres posibles casos: :@0.073088:0.290666:0.269292:0.290666:0.269292:0.274232:0.073088:0.274232:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Discriminante: b – 4ac > 0 :@0.119207:0.329798:0.315103:0.329798:0.315103:0.314178:0.119207:0.314178:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 2:@0.240132:0.324286:0.245546:0.324286:0.245546:0.315179:0.240132:0.315179:0.000000 Discriminante positivo :@0.132068:0.344927:0.302235:0.344927:0.302235:0.329307:0.132068:0.329307:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Dos raíces reales diferentes:@0.115267:0.360056:0.315663:0.360056:0.315663:0.344436:0.115267:0.344436:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Discriminante: b – 4ac = 0 :@0.403536:0.329798:0.599431:0.329798:0.599431:0.314178:0.403536:0.314178:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 2:@0.524460:0.324286:0.529875:0.324286:0.529875:0.315179:0.524460:0.315179:0.000000 Discriminante nulo :@0.429633:0.344927:0.573327:0.344927:0.573327:0.329307:0.429633:0.329307:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Raíces reales iguales:@0.424479:0.360056:0.575114:0.360056:0.575114:0.344436:0.424479:0.344436:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Discriminante: b – 4ac < 0 :@0.689522:0.329798:0.882103:0.329798:0.882103:0.314178:0.689522:0.314178:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 2:@0.810445:0.324286:0.815860:0.324286:0.815860:0.315179:0.810445:0.315179:0.000000 Discriminante negativo :@0.697947:0.344927:0.873686:0.344927:0.873686:0.329307:0.697947:0.329307:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 No hay raíces reales:@0.711702:0.360056:0.856562:0.360056:0.856562:0.344436:0.711702:0.344436:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Ejemplo::@0.079999:0.382018:0.137329:0.382018:0.137329:0.367079:0.079999:0.367079:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 x + x :@0.137212:0.382018:0.180077:0.382021:0.180077:0.367106:0.137212:0.367104:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 2:@0.149822:0.376508:0.154583:0.376508:0.154583:0.367798:0.149822:0.367798:0.000000 – 6 = 0:@0.180010:0.382021:0.225276:0.382021:0.225276:0.367081:0.180010:0.367081:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Analizamos el discriminante.:@0.079975:0.407866:0.269081:0.407866:0.269081:0.392926:0.079975:0.392926:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 a , b :@0.079975:0.422995:0.133038:0.422995:0.133038:0.408080:0.079975:0.408080:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 = 1:@0.091405:0.422995:0.112556:0.422995:0.112556:0.408055:0.091405:0.408055:0.000000:0.000000:0.000000 = 1 = 6:@0.133038:0.422995:0.200376:0.422995:0.200376:0.408055:0.133038:0.408055:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 , c –:@0.154843:0.422995:0.192209:0.422995:0.192209:0.408080:0.154843:0.408080:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 b – ac > :@0.079975:0.438124:0.149449:0.438129:0.149449:0.423214:0.079975:0.423209:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 2:@0.091300:0.432616:0.096061:0.432616:0.096061:0.423907:0.091300:0.423907:0.000000 4:@0.110225:0.438129:0.118391:0.438129:0.118391:0.423189:0.110225:0.423189:0.000000 0:@0.149399:0.438129:0.157565:0.438129:0.157565:0.423189:0.149399:0.423189:0.000000 1 4(1)( 6) 0:@0.080004:0.453258:0.182088:0.453258:0.182088:0.438318:0.080004:0.438318:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 2:@0.089165:0.447745:0.093926:0.447745:0.093926:0.439036:0.089165:0.439036:0.000000 – :@0.093884:0.453258:0.108141:0.453258:0.108141:0.438343:0.093884:0.438343:0.000000:0.000000:0.000000 – > :@0.137358:0.453258:0.173972:0.453258:0.173972:0.438343:0.137358:0.438343:0.000000:0.000000:0.000000:0.000000:0.000000 1 + 24 0:@0.079978:0.468387:0.144035:0.468387:0.144035:0.453447:0.079978:0.453447:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 > :@0.088077:0.468387:0.135919:0.468387:0.135919:0.453472:0.088077:0.453472:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 25 0:@0.079978:0.483516:0.120139:0.483516:0.120139:0.468576:0.079978:0.468576:0.000000:0.000000:0.000000:0.000000 > :@0.096159:0.483516:0.112023:0.483516:0.112023:0.468601:0.096159:0.468601:0.000000:0.000000:0.000000 La ecuación tiene dos raíces reales :@0.079978:0.505793:0.312291:0.505793:0.312291:0.490853:0.079978:0.490853:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 diferentes.:@0.079978:0.520922:0.148819:0.520922:0.148819:0.505982:0.079978:0.505982:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Graficamos la función cuadrática::@0.079978:0.546767:0.300262:0.546767:0.300262:0.531828:0.079978:0.531828:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 y x x :@0.079978:0.561896:0.153679:0.561900:0.153679:0.546986:0.079978:0.546982:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 =:@0.089650:0.561896:0.099623:0.561896:0.099623:0.546957:0.089650:0.546957:0.000000 2:@0.112197:0.556388:0.116958:0.556388:0.116958:0.547678:0.112197:0.547678:0.000000 + – 6:@0.119860:0.561900:0.161779:0.561900:0.161779:0.546960:0.119860:0.546960:0.000000:0.000000:0.000000:0.000000:0.000000 0:@0.202280:0.733398:0.208465:0.733398:0.208465:0.721466:0.202280:0.721466:0.000000 x:@0.319318:0.728670:0.325418:0.728670:0.325418:0.715247:0.319318:0.715247:0.000000 y:@0.196560:0.588051:0.202675:0.588051:0.202675:0.574627:0.196560:0.574627:0.000000 y = x + x – 6:@0.246887:0.648168:0.310064:0.648168:0.310064:0.636236:0.246887:0.636236:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 2:@0.270541:0.643758:0.274146:0.643758:0.274146:0.636802:0.270541:0.636802:0.000000 7:@0.200565:0.602385:0.205976:0.602385:0.205976:0.591945:0.200565:0.591945:0.000000 6:@0.200565:0.620254:0.205976:0.620254:0.205976:0.609814:0.200565:0.609814:0.000000 5:@0.200565:0.638125:0.205976:0.638125:0.205976:0.627685:0.200565:0.627685:0.000000 4:@0.200565:0.655996:0.205976:0.655996:0.205976:0.645556:0.200565:0.645556:0.000000 3:@0.200565:0.673867:0.205976:0.673867:0.205976:0.663427:0.200565:0.663427:0.000000 2:@0.200565:0.691738:0.205976:0.691738:0.205976:0.681298:0.200565:0.681298:0.000000 1:@0.200565:0.709609:0.205976:0.709609:0.205976:0.699169:0.200565:0.699169:0.000000 1:@0.233047:0.735793:0.238458:0.735793:0.238458:0.725353:0.233047:0.725353:0.000000 2:@0.256767:0.735793:0.262179:0.735793:0.262179:0.725353:0.256767:0.725353:0.000000 3:@0.280487:0.735793:0.285899:0.735793:0.285899:0.725353:0.280487:0.725353:0.000000 4:@0.304207:0.735793:0.309619:0.735793:0.309619:0.725353:0.304207:0.725353:0.000000 –1:@0.194719:0.745351:0.205988:0.745351:0.205988:0.734911:0.194719:0.734911:0.000000:0.000000 –1:@0.183381:0.735793:0.194649:0.735793:0.194649:0.725353:0.183381:0.725353:0.000000:0.000000 –2:@0.159660:0.735793:0.170929:0.735793:0.170929:0.725353:0.159660:0.725353:0.000000:0.000000 –3:@0.135940:0.735793:0.147209:0.735793:0.147209:0.725353:0.135940:0.725353:0.000000:0.000000 –4:@0.112220:0.735793:0.123488:0.735793:0.123488:0.725353:0.112220:0.725353:0.000000:0.000000 –2:@0.194719:0.763222:0.205988:0.763222:0.205988:0.752782:0.194719:0.752782:0.000000:0.000000 –3:@0.194719:0.781093:0.205988:0.781093:0.205988:0.770653:0.194719:0.770653:0.000000:0.000000 –4:@0.194719:0.798964:0.205988:0.798964:0.205988:0.788524:0.194719:0.788524:0.000000:0.000000 –5:@0.194719:0.816835:0.205988:0.816835:0.205988:0.806395:0.194719:0.806395:0.000000:0.000000 –6:@0.194719:0.834706:0.205988:0.834706:0.205988:0.824266:0.194719:0.824266:0.000000:0.000000 Los puntos (–3, 0) y (2, 0) son los cortes :@0.079992:0.875555:0.339449:0.875555:0.339449:0.860615:0.079992:0.860615:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 de la parábola con el eje , y son las :@0.079992:0.890684:0.314416:0.890684:0.314416:0.875744:0.079992:0.875744:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 x :@0.240302:0.890684:0.266055:0.890684:0.266055:0.875769:0.240302:0.875769:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 soluciones de la ecuación::@0.079975:0.905813:0.252233:0.905813:0.252233:0.890873:0.079975:0.890873:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 x :@0.079975:0.924509:0.094193:0.924515:0.094193:0.909601:0.079975:0.909595:0.000000:0.000000:0.000000 1:@0.086708:0.927399:0.091216:0.927399:0.091216:0.918704:0.086708:0.918704:0.000000 = –3 :@0.094126:0.924515:0.127694:0.924515:0.127694:0.909575:0.094126:0.909575:0.000000:0.000000:0.000000:0.000000:0.000000 y x :@0.127593:0.924515:0.163295:0.924515:0.163295:0.909601:0.127593:0.909601:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 2:@0.155810:0.927399:0.160317:0.927399:0.160317:0.918704:0.155810:0.918704:0.000000 = 2.:@0.163228:0.924515:0.187743:0.924515:0.187743:0.909575:0.163228:0.909575:0.000000:0.000000:0.000000:0.000000 Ejemplo: :@0.364319:0.382017:0.425247:0.382017:0.425247:0.367077:0.364319:0.367077:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 x – x + = :@0.425130:0.382017:0.508302:0.382021:0.508302:0.367106:0.425130:0.367102:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 2:@0.434792:0.376508:0.439553:0.376508:0.439553:0.367798:0.434792:0.367798:0.000000 6:@0.453716:0.382021:0.461882:0.382021:0.461882:0.367081:0.453716:0.367081:0.000000 9 0:@0.484339:0.382021:0.516418:0.382021:0.516418:0.367081:0.484339:0.367081:0.000000:0.000000:0.000000 Analizamos el discriminante:@0.364324:0.407866:0.548935:0.407866:0.548935:0.392926:0.364324:0.392926:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 .:@0.548714:0.407866:0.551743:0.407866:0.551743:0.392952:0.548714:0.392952:0.000000 a :@0.364324:0.422995:0.375686:0.422995:0.375686:0.408080:0.364324:0.408080:0.000000:0.000000 = 1:@0.375619:0.422995:0.397290:0.422995:0.397290:0.408055:0.375619:0.408055:0.000000:0.000000:0.000000 , b – , c :@0.397206:0.422995:0.478265:0.422995:0.478265:0.408080:0.397206:0.408080:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 = 6:@0.417370:0.422995:0.446705:0.422995:0.446705:0.408055:0.417370:0.408055:0.000000:0.000000:0.000000 = 9:@0.465346:0.422995:0.486364:0.422995:0.486364:0.408055:0.465346:0.408055:0.000000:0.000000:0.000000 b – ac :@0.364290:0.438124:0.433778:0.438129:0.433778:0.423214:0.364290:0.423209:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 2:@0.375629:0.432616:0.380390:0.432616:0.380390:0.423907:0.375629:0.423907:0.000000 4:@0.394554:0.438129:0.402720:0.438129:0.402720:0.423189:0.394554:0.423189:0.000000 = 0:@0.420859:0.438129:0.441877:0.438129:0.441877:0.423189:0.420859:0.423189:0.000000:0.000000:0.000000 ( 6):@0.364316:0.453258:0.389517:0.453258:0.389517:0.438318:0.364316:0.438318:0.000000:0.000000:0.000000:0.000000 –:@0.368683:0.453258:0.377050:0.453258:0.377050:0.438343:0.368683:0.438343:0.000000 2:@0.389441:0.447745:0.394202:0.447745:0.394202:0.439036:0.389441:0.439036:0.000000 – :@0.394160:0.453258:0.408418:0.453258:0.408418:0.438343:0.394160:0.438343:0.000000:0.000000:0.000000 4(1)(9) = 0:@0.408367:0.453258:0.475403:0.453258:0.475403:0.438318:0.408367:0.438318:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 36 36 = 0:@0.364324:0.468387:0.436179:0.468387:0.436179:0.453447:0.364324:0.453447:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 – :@0.380506:0.468387:0.394763:0.468387:0.394763:0.453472:0.380506:0.453472:0.000000:0.000000:0.000000 0 = 0:@0.364324:0.483516:0.397691:0.483516:0.397691:0.468576:0.364324:0.468576:0.000000:0.000000:0.000000:0.000000:0.000000 La ecuación tiene única solución, :@0.364324:0.505793:0.585897:0.505793:0.585897:0.490853:0.364324:0.490853:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 las dos raíces son reales e iguales.:@0.364324:0.520922:0.583758:0.520922:0.583758:0.505982:0.364324:0.505982:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Graficamos la función cuadrática::@0.364324:0.546767:0.582462:0.546767:0.582462:0.531828:0.364324:0.531828:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 y x – x + :@0.364324:0.561896:0.446124:0.561900:0.446124:0.546986:0.364324:0.546982:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 =:@0.373996:0.561896:0.383969:0.561896:0.383969:0.546957:0.373996:0.546957:0.000000 2:@0.396527:0.556388:0.401287:0.556388:0.401287:0.547678:0.396527:0.547678:0.000000 6:@0.415451:0.561900:0.423617:0.561900:0.423617:0.546960:0.415451:0.546960:0.000000 9 :@0.446074:0.561900:0.457905:0.561900:0.457905:0.546960:0.446074:0.546960:0.000000:0.000000 0:@0.439174:0.804873:0.445359:0.804873:0.445359:0.792942:0.439174:0.792942:0.000000 y:@0.434171:0.588051:0.440286:0.588051:0.440286:0.574627:0.434171:0.574627:0.000000 x:@0.603647:0.800143:0.609746:0.800143:0.609746:0.786720:0.603647:0.786720:0.000000 y = x – 6x + 9:@0.524537:0.633039:0.593899:0.633039:0.593899:0.621107:0.524537:0.621107:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 2:@0.548191:0.628629:0.551796:0.628629:0.551796:0.621673:0.548191:0.621673:0.000000 1:@0.437036:0.781345:0.442448:0.781345:0.442448:0.770905:0.437036:0.770905:0.000000 1:@0.469941:0.807314:0.475353:0.807314:0.475353:0.796874:0.469941:0.796874:0.000000 2:@0.493662:0.807314:0.499073:0.807314:0.499073:0.796874:0.493662:0.796874:0.000000 3:@0.517382:0.807314:0.522794:0.807314:0.522794:0.796874:0.517382:0.796874:0.000000 4:@0.541102:0.807314:0.546514:0.807314:0.546514:0.796874:0.541102:0.796874:0.000000 5:@0.564822:0.807314:0.570234:0.807314:0.570234:0.796874:0.564822:0.796874:0.000000 2:@0.437049:0.763479:0.442461:0.763479:0.442461:0.753039:0.437049:0.753039:0.000000 3:@0.437049:0.745608:0.442461:0.745608:0.442461:0.735168:0.437049:0.735168:0.000000 4:@0.437049:0.727737:0.442461:0.727737:0.442461:0.717297:0.437049:0.717297:0.000000 5:@0.437049:0.709866:0.442461:0.709866:0.442461:0.699426:0.437049:0.699426:0.000000 6:@0.437049:0.691995:0.442461:0.691995:0.442461:0.681555:0.437049:0.681555:0.000000 7:@0.437049:0.674124:0.442461:0.674124:0.442461:0.663684:0.437049:0.663684:0.000000 8:@0.437049:0.656253:0.442461:0.656253:0.442461:0.645813:0.437049:0.645813:0.000000 9:@0.437049:0.638382:0.442461:0.638382:0.442461:0.627942:0.437049:0.627942:0.000000 10:@0.431638:0.620511:0.442461:0.620511:0.442461:0.610071:0.431638:0.610071:0.000000:0.000000 –1:@0.431192:0.817066:0.442461:0.817066:0.442461:0.806626:0.431192:0.806626:0.000000:0.000000 –1:@0.419514:0.807297:0.430782:0.807297:0.430782:0.796856:0.419514:0.796856:0.000000:0.000000 –2:@0.395794:0.807297:0.407062:0.807297:0.407062:0.796856:0.395794:0.796856:0.000000:0.000000 –2:@0.431192:0.834937:0.442461:0.834937:0.442461:0.824497:0.431192:0.824497:0.000000:0.000000 El punto de corte de la parábola con el :@0.364321:0.875555:0.623252:0.875555:0.623252:0.860615:0.364321:0.860615:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 eje es (3, 0), que es el vértice de la :@0.364321:0.890684:0.598297:0.890684:0.598297:0.875744:0.364321:0.875744:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 x :@0.383916:0.890684:0.396583:0.890684:0.396583:0.875769:0.383916:0.875769:0.000000:0.000000:0.000000 parábola. La ecuación tiene dos raíces :@0.364304:0.905813:0.617328:0.905813:0.617328:0.890873:0.364304:0.890873:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 reales iguales::@0.364304:0.920941:0.455296:0.920941:0.455296:0.906002:0.364304:0.906002:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 x x :@0.455135:0.920941:0.502251:0.920941:0.502251:0.906027:0.455135:0.906027:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 1:@0.467774:0.923826:0.472282:0.923826:0.472282:0.915130:0.467774:0.915130:0.000000 =:@0.475192:0.920941:0.485166:0.920941:0.485166:0.906002:0.475192:0.906002:0.000000 2:@0.494768:0.923826:0.499275:0.923826:0.499275:0.915130:0.494768:0.915130:0.000000 = 3.:@0.502184:0.920941:0.526699:0.920941:0.526699:0.906002:0.502184:0.906002:0.000000:0.000000:0.000000:0.000000 Ejemplo:@0.648639:0.382023:0.703124:0.382023:0.703124:0.367084:0.648639:0.367084:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 : x :@0.703007:0.382023:0.726341:0.382021:0.726341:0.367106:0.703007:0.367109:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 2:@0.718610:0.376508:0.723371:0.376508:0.723371:0.367798:0.718610:0.367798:0.000000 – 3 + 5 = 0:@0.726274:0.382021:0.802781:0.382021:0.802781:0.367081:0.726274:0.367081:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 x :@0.746254:0.382021:0.755977:0.382021:0.755977:0.367106:0.746254:0.367106:0.000000:0.000000 Analizamos el discriminante.:@0.648629:0.407866:0.835911:0.407866:0.835911:0.392926:0.648629:0.392926:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 a = b :@0.648629:0.422995:0.698680:0.422995:0.698680:0.408080:0.648629:0.408080:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 1, = –3, = 5:@0.672793:0.422995:0.766435:0.422995:0.766435:0.408055:0.672793:0.408055:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 c :@0.734926:0.422995:0.744832:0.422995:0.744832:0.408080:0.734926:0.408080:0.000000:0.000000 b – ac :@0.648595:0.438124:0.705891:0.438129:0.705891:0.423214:0.648595:0.423209:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 2:@0.659958:0.432616:0.664719:0.432616:0.664719:0.423907:0.659958:0.423907:0.000000 4:@0.675937:0.438129:0.687701:0.438129:0.687701:0.423189:0.675937:0.423189:0.000000:0.000000 < 0:@0.705824:0.438129:0.727494:0.438129:0.727494:0.423189:0.705824:0.423189:0.000000:0.000000:0.000000 (–3):@0.648644:0.453258:0.673845:0.453258:0.673845:0.438318:0.648644:0.438318:0.000000:0.000000:0.000000:0.000000 2:@0.673758:0.447745:0.678519:0.447745:0.678519:0.439036:0.673758:0.439036:0.000000 – :@0.678475:0.453258:0.692733:0.453258:0.692733:0.438343:0.678475:0.438343:0.000000:0.000000:0.000000 4(1)(5) < 0:@0.692683:0.453258:0.759718:0.453258:0.759718:0.438318:0.692683:0.438318:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 9 – 20 < 0:@0.648639:0.468387:0.713700:0.468387:0.713700:0.453447:0.648639:0.453447:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 –11 < 0:@0.648639:0.483516:0.698406:0.483516:0.698406:0.468576:0.648639:0.468576:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 La ecuación no tiene solución real.:@0.648639:0.505793:0.875554:0.505793:0.875554:0.490853:0.648639:0.490853:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Graficamos la función cuadrática::@0.648639:0.531639:0.866698:0.531639:0.866698:0.516699:0.648639:0.516699:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 y x :@0.648639:0.546767:0.688584:0.546771:0.688584:0.531857:0.648639:0.531853:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 =:@0.658311:0.546767:0.668285:0.546767:0.668285:0.531828:0.658311:0.531828:0.000000 2:@0.680855:0.541259:0.685616:0.541259:0.685616:0.532549:0.680855:0.532549:0.000000 – 3 + 5 :@0.688517:0.546771:0.743488:0.546771:0.743488:0.531831:0.688517:0.531831:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 x :@0.708498:0.546771:0.718220:0.546771:0.718220:0.531857:0.708498:0.531857:0.000000:0.000000 0:@0.723503:0.789745:0.729688:0.789745:0.729688:0.777813:0.723503:0.777813:0.000000 y:@0.717414:0.572922:0.723528:0.572922:0.723528:0.559498:0.717414:0.559498:0.000000 x:@0.878636:0.795431:0.884735:0.795431:0.884735:0.782007:0.878636:0.782007:0.000000 y = x – 3x + 5:@0.806493:0.617910:0.875856:0.617910:0.875856:0.605978:0.806493:0.605978:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 2:@0.830148:0.613500:0.833754:0.613500:0.833754:0.606544:0.830148:0.606544:0.000000 1:@0.721128:0.766217:0.726540:0.766217:0.726540:0.755776:0.721128:0.755776:0.000000 1:@0.755061:0.792185:0.760473:0.792185:0.760473:0.781745:0.755061:0.781745:0.000000 2:@0.778782:0.792185:0.784193:0.792185:0.784193:0.781745:0.778782:0.781745:0.000000 3:@0.802502:0.792185:0.807914:0.792185:0.807914:0.781745:0.802502:0.781745:0.000000 4:@0.826222:0.792185:0.831634:0.792185:0.831634:0.781745:0.826222:0.781745:0.000000 5:@0.849942:0.792185:0.855354:0.792185:0.855354:0.781745:0.849942:0.781745:0.000000 2:@0.721138:0.748351:0.726550:0.748351:0.726550:0.737910:0.721138:0.737910:0.000000 3:@0.721138:0.730479:0.726550:0.730479:0.726550:0.720039:0.721138:0.720039:0.000000 4:@0.721138:0.712608:0.726550:0.712608:0.726550:0.702168:0.721138:0.702168:0.000000 5:@0.721138:0.694737:0.726550:0.694737:0.726550:0.684297:0.721138:0.684297:0.000000 6:@0.721138:0.676866:0.726550:0.676866:0.726550:0.666426:0.721138:0.666426:0.000000 7:@0.721138:0.658995:0.726550:0.658995:0.726550:0.648555:0.721138:0.648555:0.000000 8:@0.721138:0.641124:0.726550:0.641124:0.726550:0.630684:0.721138:0.630684:0.000000 9:@0.721138:0.623253:0.726550:0.623253:0.726550:0.612813:0.721138:0.612813:0.000000 10:@0.715727:0.605382:0.726550:0.605382:0.726550:0.594942:0.715727:0.594942:0.000000:0.000000 –1:@0.715282:0.801937:0.726550:0.801937:0.726550:0.791497:0.715282:0.791497:0.000000:0.000000 –1:@0.704634:0.792168:0.715902:0.792168:0.715902:0.781727:0.704634:0.781727:0.000000:0.000000 –2:@0.680914:0.792168:0.692182:0.792168:0.692182:0.781727:0.680914:0.781727:0.000000:0.000000 –2:@0.715282:0.819808:0.726550:0.819808:0.726550:0.809368:0.715282:0.809368:0.000000:0.000000 La parábola no corta el eje de las , :@0.648649:0.860426:0.880882:0.860426:0.880882:0.845486:0.648649:0.845486:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 x:@0.867595:0.860426:0.874372:0.860426:0.874372:0.845511:0.867595:0.845511:0.000000 por lo tanto, la ecuación no tiene :@0.648632:0.875555:0.869692:0.875555:0.869692:0.860615:0.648632:0.860615:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 soluciones reales.:@0.648632:0.890684:0.763012:0.890684:0.763012:0.875744:0.648632:0.875744:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 La identidad de Legendre :@0.740434:0.216306:0.913499:0.216306:0.913499:0.201366:0.740434:0.201366:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 se aplica a las raíces::@0.740434:0.231435:0.872665:0.231435:0.872665:0.216495:0.740434:0.216495:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Competencia :@0.740434:0.175599:0.848995:0.175599:0.848995:0.158652:0.740434:0.158652:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 matemática :@0.740434:0.192241:0.837910:0.192241:0.837910:0.175294:0.740434:0.175294:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 MUESTRA EDITORIAL:@0.248902:0.947499:0.888171:0.113285:0.743922:0.050540:0.104653:0.884754:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000 Muestra editorial solo para fines didácticos – Prohibida su venta:@0.095585:0.659700:0.095585:0.341447:0.077558:0.341447:0.077558:0.659700:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000:0.000000