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Whakaoti mō x (complex solution)
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-2x^{2}+x=8
Ko ngā whārite katoa o te āhua ax^{2}+bx+c=0 ka taea te whakaoti mā te whakamahi i te tikanga tātai pūrua: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. E rua ngā otinga ka puta i te tikanga tātai pūrua, ko tētahi ina he tāpiri a ±, ā, ko tētahi ina he tango.
-2x^{2}+x-8=8-8
Me tango 8 mai i ngā taha e rua o te whārite.
-2x^{2}+x-8=0
Mā te tango i te 8 i a ia ake anō ka toe ko te 0.
x=\frac{-1±\sqrt{1^{2}-4\left(-2\right)\left(-8\right)}}{2\left(-2\right)}
Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -2 mō a, 1 mō b, me -8 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-1±\sqrt{1-4\left(-2\right)\left(-8\right)}}{2\left(-2\right)}
Pūrua 1.
x=\frac{-1±\sqrt{1+8\left(-8\right)}}{2\left(-2\right)}
Whakareatia -4 ki te -2.
x=\frac{-1±\sqrt{1-64}}{2\left(-2\right)}
Whakareatia 8 ki te -8.
x=\frac{-1±\sqrt{-63}}{2\left(-2\right)}
Tāpiri 1 ki te -64.
x=\frac{-1±3\sqrt{7}i}{2\left(-2\right)}
Tuhia te pūtakerua o te -63.
x=\frac{-1±3\sqrt{7}i}{-4}
Whakareatia 2 ki te -2.
x=\frac{-1+3\sqrt{7}i}{-4}
Nā, me whakaoti te whārite x=\frac{-1±3\sqrt{7}i}{-4} ina he tāpiri te ±. Tāpiri -1 ki te 3i\sqrt{7}.
x=\frac{-3\sqrt{7}i+1}{4}
Whakawehe -1+3i\sqrt{7} ki te -4.
x=\frac{-3\sqrt{7}i-1}{-4}
Nā, me whakaoti te whārite x=\frac{-1±3\sqrt{7}i}{-4} ina he tango te ±. Tango 3i\sqrt{7} mai i -1.
x=\frac{1+3\sqrt{7}i}{4}
Whakawehe -1-3i\sqrt{7} ki te -4.
x=\frac{-3\sqrt{7}i+1}{4} x=\frac{1+3\sqrt{7}i}{4}
Kua oti te whārite te whakatau.
-2x^{2}+x=8
Ko ngā whārite pūrua pēnei i tēnei nā ka taea te whakaoti mā te whakaoti i te pūrua. Hei whakaoti i te pūrua, ko te whārite me mātua tuhi ki te āhua x^{2}+bx=c.
\frac{-2x^{2}+x}{-2}=\frac{8}{-2}
Whakawehea ngā taha e rua ki te -2.
x^{2}+\frac{1}{-2}x=\frac{8}{-2}
Mā te whakawehe ki te -2 ka wetekia te whakareanga ki te -2.
x^{2}-\frac{1}{2}x=\frac{8}{-2}
Whakawehe 1 ki te -2.
x^{2}-\frac{1}{2}x=-4
Whakawehe 8 ki te -2.
x^{2}-\frac{1}{2}x+\left(-\frac{1}{4}\right)^{2}=-4+\left(-\frac{1}{4}\right)^{2}
Whakawehea te -\frac{1}{2}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{1}{4}. Nā, tāpiria te pūrua o te -\frac{1}{4} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.
x^{2}-\frac{1}{2}x+\frac{1}{16}=-4+\frac{1}{16}
Pūruatia -\frac{1}{4} mā te pūrua i te taurunga me te tauraro o te hautanga.
x^{2}-\frac{1}{2}x+\frac{1}{16}=-\frac{63}{16}
Tāpiri -4 ki te \frac{1}{16}.
\left(x-\frac{1}{4}\right)^{2}=-\frac{63}{16}
Tauwehea x^{2}-\frac{1}{2}x+\frac{1}{16}. Ko te tikanga pūnoa, ina ko x^{2}+bx+c he pūrua tika pūrua tika pū, ka taea taua mea te tauwehea i ngā wā katoa hei \left(x+\frac{b}{2}\right)^{2}.
\sqrt{\left(x-\frac{1}{4}\right)^{2}}=\sqrt{-\frac{63}{16}}
Tuhia te pūtakerua o ngā taha e rua o te whārite.
x-\frac{1}{4}=\frac{3\sqrt{7}i}{4} x-\frac{1}{4}=-\frac{3\sqrt{7}i}{4}
Whakarūnātia.
x=\frac{1+3\sqrt{7}i}{4} x=\frac{-3\sqrt{7}i+1}{4}
Me tāpiri \frac{1}{4} ki ngā taha e rua o te whārite.