Whakaoti i te 3x^2-2x+9=0

$$3 { x }^{ 2 } -2x+9 = 0$$

Whakaoti mō x (complex solution)

$$x=\frac{1+\sqrt{26}i}{3}\approx 0.333333333+1.699673171i$$

$$x=\frac{-\sqrt{26}i+1}{3}\approx 0.333333333-1.699673171i$$

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.

$$3x^{2}-2x+9=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 3 mō a, -2 mō b, me 9 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\times 3\times 9}}{2\times 3}$$

Pūrua -2.

$$x=\frac{-\left(-2\right)±\sqrt{4-4\times 3\times 9}}{2\times 3}$$

Whakareatia -4 ki te 3.

$$x=\frac{-\left(-2\right)±\sqrt{4-12\times 9}}{2\times 3}$$

Whakareatia -12 ki te 9.

$$x=\frac{-\left(-2\right)±\sqrt{4-108}}{2\times 3}$$

Tāpiri 4 ki te -108.

$$x=\frac{-\left(-2\right)±\sqrt{-104}}{2\times 3}$$

Tuhia te pūtakerua o te -104.

$$x=\frac{-\left(-2\right)±2\sqrt{26}i}{2\times 3}$$

Ko te tauaro o -2 ko 2.

$$x=\frac{2±2\sqrt{26}i}{2\times 3}$$

Whakareatia 2 ki te 3.

$$x=\frac{2±2\sqrt{26}i}{6}$$

Nā, me whakaoti te whārite x=\frac{2±2\sqrt{26}i}{6} ina he tāpiri te ±. Tāpiri 2 ki te 2i\sqrt{26}.

$$x=\frac{2+2\sqrt{26}i}{6}$$

Whakawehe 2+2i\sqrt{26} ki te 6.

$$x=\frac{1+\sqrt{26}i}{3}$$

Nā, me whakaoti te whārite x=\frac{2±2\sqrt{26}i}{6} ina he tango te ±. Tango 2i\sqrt{26} mai i 2.

$$x=\frac{-2\sqrt{26}i+2}{6}$$

Whakawehe 2-2i\sqrt{26} ki te 6.

$$x=\frac{-\sqrt{26}i+1}{3}$$

Kua oti te whārite te whakatau.

$$x=\frac{1+\sqrt{26}i}{3} x=\frac{-\sqrt{26}i+1}{3}$$

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.

$$3x^{2}-2x+9=0$$

Me tango 9 mai i ngā taha e rua o te whārite.

$$3x^{2}-2x+9-9=-9$$

Mā te tango i te 9 i a ia ake anō ka toe ko te 0.

$$3x^{2}-2x=-9$$

Whakawehea ngā taha e rua ki te 3.

$$\frac{3x^{2}-2x}{3}=-\frac{9}{3}$$

Mā te whakawehe ki te 3 ka wetekia te whakareanga ki te 3.

$$x^{2}-\frac{2}{3}x=-\frac{9}{3}$$

Whakawehe -9 ki te 3.

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

Whakawehea te -\frac{2}{3}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{1}{3}. Nā, tāpiria te pūrua o te -\frac{1}{3} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{2}{3}x+\left(-\frac{1}{3}\right)^{2}=-3+\left(-\frac{1}{3}\right)^{2}$$

Pūruatia -\frac{1}{3} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{2}{3}x+\frac{1}{9}=-3+\frac{1}{9}$$

Tāpiri -3 ki te \frac{1}{9}.

$$x^{2}-\frac{2}{3}x+\frac{1}{9}=-\frac{26}{9}$$

Tauwehea x^{2}-\frac{2}{3}x+\frac{1}{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}.

$$\left(x-\frac{1}{3}\right)^{2}=-\frac{26}{9}$$

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

$$\sqrt{\left(x-\frac{1}{3}\right)^{2}}=\sqrt{-\frac{26}{9}}$$

Whakarūnātia.

$$x-\frac{1}{3}=\frac{\sqrt{26}i}{3} x-\frac{1}{3}=-\frac{\sqrt{26}i}{3}$$

Me tāpiri \frac{1}{3} ki ngā taha e rua o te whārite.

$$x=\frac{1+\sqrt{26}i}{3} x=\frac{-\sqrt{26}i+1}{3}$$