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Whakaoti mō x (complex solution)
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4x^{2}-11x+30=16
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.
4x^{2}-11x+30-16=16-16
Me tango 16 mai i ngā taha e rua o te whārite.
4x^{2}-11x+30-16=0
Mā te tango i te 16 i a ia ake anō ka toe ko te 0.
4x^{2}-11x+14=0
Tango 16 mai i 30.
x=\frac{-\left(-11\right)±\sqrt{\left(-11\right)^{2}-4\times 4\times 14}}{2\times 4}
Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 4 mō a, -11 mō b, me 14 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-11\right)±\sqrt{121-4\times 4\times 14}}{2\times 4}
Pūrua -11.
x=\frac{-\left(-11\right)±\sqrt{121-16\times 14}}{2\times 4}
Whakareatia -4 ki te 4.
x=\frac{-\left(-11\right)±\sqrt{121-224}}{2\times 4}
Whakareatia -16 ki te 14.
x=\frac{-\left(-11\right)±\sqrt{-103}}{2\times 4}
Tāpiri 121 ki te -224.
x=\frac{-\left(-11\right)±\sqrt{103}i}{2\times 4}
Tuhia te pūtakerua o te -103.
x=\frac{11±\sqrt{103}i}{2\times 4}
Ko te tauaro o -11 ko 11.
x=\frac{11±\sqrt{103}i}{8}
Whakareatia 2 ki te 4.
x=\frac{11+\sqrt{103}i}{8}
Nā, me whakaoti te whārite x=\frac{11±\sqrt{103}i}{8} ina he tāpiri te ±. Tāpiri 11 ki te i\sqrt{103}.
x=\frac{-\sqrt{103}i+11}{8}
Nā, me whakaoti te whārite x=\frac{11±\sqrt{103}i}{8} ina he tango te ±. Tango i\sqrt{103} mai i 11.
x=\frac{11+\sqrt{103}i}{8} x=\frac{-\sqrt{103}i+11}{8}
Kua oti te whārite te whakatau.
4x^{2}-11x+30=16
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.
4x^{2}-11x+30-30=16-30
Me tango 30 mai i ngā taha e rua o te whārite.
4x^{2}-11x=16-30
Mā te tango i te 30 i a ia ake anō ka toe ko te 0.
4x^{2}-11x=-14
Tango 30 mai i 16.
\frac{4x^{2}-11x}{4}=-\frac{14}{4}
Whakawehea ngā taha e rua ki te 4.
x^{2}-\frac{11}{4}x=-\frac{14}{4}
Mā te whakawehe ki te 4 ka wetekia te whakareanga ki te 4.
x^{2}-\frac{11}{4}x=-\frac{7}{2}
Whakahekea te hautanga \frac{-14}{4} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.
x^{2}-\frac{11}{4}x+\left(-\frac{11}{8}\right)^{2}=-\frac{7}{2}+\left(-\frac{11}{8}\right)^{2}
Whakawehea te -\frac{11}{4}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{11}{8}. Nā, tāpiria te pūrua o te -\frac{11}{8} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.
x^{2}-\frac{11}{4}x+\frac{121}{64}=-\frac{7}{2}+\frac{121}{64}
Pūruatia -\frac{11}{8} mā te pūrua i te taurunga me te tauraro o te hautanga.
x^{2}-\frac{11}{4}x+\frac{121}{64}=-\frac{103}{64}
Tāpiri -\frac{7}{2} ki te \frac{121}{64} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.
\left(x-\frac{11}{8}\right)^{2}=-\frac{103}{64}
Tauwehea x^{2}-\frac{11}{4}x+\frac{121}{64}. 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{11}{8}\right)^{2}}=\sqrt{-\frac{103}{64}}
Tuhia te pūtakerua o ngā taha e rua o te whārite.
x-\frac{11}{8}=\frac{\sqrt{103}i}{8} x-\frac{11}{8}=-\frac{\sqrt{103}i}{8}
Whakarūnātia.
x=\frac{11+\sqrt{103}i}{8} x=\frac{-\sqrt{103}i+11}{8}
Me tāpiri \frac{11}{8} ki ngā taha e rua o te whārite.