Whakaoti i te 9x^2-6x+3=0

$$9 x ^ { 2 } - 6 x + 3 = 0$$

Whakaoti mō x (complex solution)

$$x=\frac{1+\sqrt{2}i}{3}\approx 0.333333333+0.471404521i$$

$$x=\frac{-\sqrt{2}i+1}{3}\approx 0.333333333-0.471404521i$$

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.

$$9x^{2}-6x+3=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 9 mō a, -6 mō b, me 3 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\times 9\times 3}}{2\times 9}$$

Pūrua -6.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\times 9\times 3}}{2\times 9}$$

Whakareatia -4 ki te 9.

$$x=\frac{-\left(-6\right)±\sqrt{36-36\times 3}}{2\times 9}$$

Whakareatia -36 ki te 3.

$$x=\frac{-\left(-6\right)±\sqrt{36-108}}{2\times 9}$$

Tāpiri 36 ki te -108.

$$x=\frac{-\left(-6\right)±\sqrt{-72}}{2\times 9}$$

Tuhia te pūtakerua o te -72.

$$x=\frac{-\left(-6\right)±6\sqrt{2}i}{2\times 9}$$

Ko te tauaro o -6 ko 6.

$$x=\frac{6±6\sqrt{2}i}{2\times 9}$$

Whakareatia 2 ki te 9.

$$x=\frac{6±6\sqrt{2}i}{18}$$

Nā, me whakaoti te whārite x=\frac{6±6\sqrt{2}i}{18} ina he tāpiri te ±. Tāpiri 6 ki te 6i\sqrt{2}.

$$x=\frac{6+6\sqrt{2}i}{18}$$

Whakawehe 6+6i\sqrt{2} ki te 18.

$$x=\frac{1+\sqrt{2}i}{3}$$

Nā, me whakaoti te whārite x=\frac{6±6\sqrt{2}i}{18} ina he tango te ±. Tango 6i\sqrt{2} mai i 6.

$$x=\frac{-6\sqrt{2}i+6}{18}$$

Whakawehe 6-6i\sqrt{2} ki te 18.

$$x=\frac{-\sqrt{2}i+1}{3}$$

Kua oti te whārite te whakatau.

$$x=\frac{1+\sqrt{2}i}{3} x=\frac{-\sqrt{2}i+1}{3}$$

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.

$$9x^{2}-6x+3=0$$

Me tango 3 mai i ngā taha e rua o te whārite.

$$9x^{2}-6x+3-3=-3$$

Mā te tango i te 3 i a ia ake anō ka toe ko te 0.

$$9x^{2}-6x=-3$$

Whakawehea ngā taha e rua ki te 9.

$$\frac{9x^{2}-6x}{9}=-\frac{3}{9}$$

Mā te whakawehe ki te 9 ka wetekia te whakareanga ki te 9.

$$x^{2}+\left(-\frac{6}{9}\right)x=-\frac{3}{9}$$

Whakahekea te hautanga \frac{-6}{9} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 3.

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

Whakahekea te hautanga \frac{-3}{9} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 3.

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

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.

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

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

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

Tāpiri -\frac{1}{3} ki te \frac{1}{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.

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

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}.

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

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

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

Whakarūnātia.

$$x-\frac{1}{3}=\frac{\sqrt{2}i}{3} x-\frac{1}{3}=-\frac{\sqrt{2}i}{3}$$

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

$$x=\frac{1+\sqrt{2}i}{3} x=\frac{-\sqrt{2}i+1}{3}$$