Whakaoti i te 12x^2-88x+400=0

$$12 { x }^{ 2 } -88x+400 = 0$$

Whakaoti mō x (complex solution)

$$x=\frac{11+\sqrt{179}i}{3}\approx 3.666666667+4.459696053i$$

$$x=\frac{-\sqrt{179}i+11}{3}\approx 3.666666667-4.459696053i$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

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.

$$12x^{2}-88x+400=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 12 mō a, -88 mō b, me 400 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-88\right)±\sqrt{\left(-88\right)^{2}-4\times 12\times 400}}{2\times 12}$$

Pūrua -88.

$$x=\frac{-\left(-88\right)±\sqrt{7744-4\times 12\times 400}}{2\times 12}$$

Whakareatia -4 ki te 12.

$$x=\frac{-\left(-88\right)±\sqrt{7744-48\times 400}}{2\times 12}$$

Whakareatia -48 ki te 400.

$$x=\frac{-\left(-88\right)±\sqrt{7744-19200}}{2\times 12}$$

Tāpiri 7744 ki te -19200.

$$x=\frac{-\left(-88\right)±\sqrt{-11456}}{2\times 12}$$

Tuhia te pūtakerua o te -11456.

$$x=\frac{-\left(-88\right)±8\sqrt{179}i}{2\times 12}$$

Ko te tauaro o -88 ko 88.

$$x=\frac{88±8\sqrt{179}i}{2\times 12}$$

Whakareatia 2 ki te 12.

$$x=\frac{88±8\sqrt{179}i}{24}$$

Nā, me whakaoti te whārite x=\frac{88±8\sqrt{179}i}{24} ina he tāpiri te ±. Tāpiri 88 ki te 8i\sqrt{179}.

$$x=\frac{88+8\sqrt{179}i}{24}$$

Whakawehe 88+8i\sqrt{179} ki te 24.

$$x=\frac{11+\sqrt{179}i}{3}$$

Nā, me whakaoti te whārite x=\frac{88±8\sqrt{179}i}{24} ina he tango te ±. Tango 8i\sqrt{179} mai i 88.

$$x=\frac{-8\sqrt{179}i+88}{24}$$

Whakawehe 88-8i\sqrt{179} ki te 24.

$$x=\frac{-\sqrt{179}i+11}{3}$$

Kua oti te whārite te whakatau.

$$x=\frac{11+\sqrt{179}i}{3} x=\frac{-\sqrt{179}i+11}{3}$$

Ngā Upane mō te Whakaoti i te Pūrua

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.

$$12x^{2}-88x+400=0$$

Me tango 400 mai i ngā taha e rua o te whārite.

$$12x^{2}-88x+400-400=-400$$

Mā te tango i te 400 i a ia ake anō ka toe ko te 0.

$$12x^{2}-88x=-400$$

Whakawehea ngā taha e rua ki te 12.

$$\frac{12x^{2}-88x}{12}=-\frac{400}{12}$$

Mā te whakawehe ki te 12 ka wetekia te whakareanga ki te 12.

$$x^{2}+\left(-\frac{88}{12}\right)x=-\frac{400}{12}$$

Whakahekea te hautanga \frac{-88}{12} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

$$x^{2}-\frac{22}{3}x=-\frac{400}{12}$$

Whakahekea te hautanga \frac{-400}{12} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

$$x^{2}-\frac{22}{3}x=-\frac{100}{3}$$

Whakawehea te -\frac{22}{3}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{11}{3}. Nā, tāpiria te pūrua o te -\frac{11}{3} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{22}{3}x+\left(-\frac{11}{3}\right)^{2}=-\frac{100}{3}+\left(-\frac{11}{3}\right)^{2}$$

Pūruatia -\frac{11}{3} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{22}{3}x+\frac{121}{9}=-\frac{100}{3}+\frac{121}{9}$$

Tāpiri -\frac{100}{3} ki te \frac{121}{9} 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.

$$x^{2}-\frac{22}{3}x+\frac{121}{9}=-\frac{179}{9}$$

Tauwehea x^{2}-\frac{22}{3}x+\frac{121}{9}. 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}.

$$\left(x-\frac{11}{3}\right)^{2}=-\frac{179}{9}$$

Tuhia te pūtakerua o ngā taha e rua o te whārite.

$$\sqrt{\left(x-\frac{11}{3}\right)^{2}}=\sqrt{-\frac{179}{9}}$$

Whakarūnātia.

$$x-\frac{11}{3}=\frac{\sqrt{179}i}{3} x-\frac{11}{3}=-\frac{\sqrt{179}i}{3}$$

Me tāpiri \frac{11}{3} ki ngā taha e rua o te whārite.

$$x=\frac{11+\sqrt{179}i}{3} x=\frac{-\sqrt{179}i+11}{3}$$