Whakaoti i te 3x^2-6x+36=0

$$3 { x }^{ 2 } -6x+36=0$$

Whakaoti mō x (complex solution)

$$x=1+\sqrt{11}i\approx 1+3.31662479i$$

$$x=-\sqrt{11}i+1\approx 1-3.31662479i$$

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.

$$3x^{2}-6x+36=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 3 mō a, -6 mō b, me 36 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

Pūrua -6.

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

Whakareatia -4 ki te 3.

$$x=\frac{-\left(-6\right)±\sqrt{36-12\times 36}}{2\times 3}$$

Whakareatia -12 ki te 36.

$$x=\frac{-\left(-6\right)±\sqrt{36-432}}{2\times 3}$$

Tāpiri 36 ki te -432.

$$x=\frac{-\left(-6\right)±\sqrt{-396}}{2\times 3}$$

Tuhia te pūtakerua o te -396.

$$x=\frac{-\left(-6\right)±6\sqrt{11}i}{2\times 3}$$

Ko te tauaro o -6 ko 6.

$$x=\frac{6±6\sqrt{11}i}{2\times 3}$$

Whakareatia 2 ki te 3.

$$x=\frac{6±6\sqrt{11}i}{6}$$

Nā, me whakaoti te whārite x=\frac{6±6\sqrt{11}i}{6} ina he tāpiri te ±. Tāpiri 6 ki te 6i\sqrt{11}.

$$x=\frac{6+6\sqrt{11}i}{6}$$

Whakawehe 6+6i\sqrt{11} ki te 6.

$$x=1+\sqrt{11}i$$

Nā, me whakaoti te whārite x=\frac{6±6\sqrt{11}i}{6} ina he tango te ±. Tango 6i\sqrt{11} mai i 6.

$$x=\frac{-6\sqrt{11}i+6}{6}$$

Whakawehe 6-6i\sqrt{11} ki te 6.

$$x=-\sqrt{11}i+1$$

Kua oti te whārite te whakatau.

$$x=1+\sqrt{11}i x=-\sqrt{11}i+1$$

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.

$$3x^{2}-6x+36=0$$

Me tango 36 mai i ngā taha e rua o te whārite.

$$3x^{2}-6x+36-36=-36$$

Mā te tango i te 36 i a ia ake anō ka toe ko te 0.

$$3x^{2}-6x=-36$$

Whakawehea ngā taha e rua ki te 3.

$$\frac{3x^{2}-6x}{3}=-\frac{36}{3}$$

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

$$x^{2}+\left(-\frac{6}{3}\right)x=-\frac{36}{3}$$

Whakawehe -6 ki te 3.

$$x^{2}-2x=-\frac{36}{3}$$

Whakawehe -36 ki te 3.

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

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

$$x^{2}-2x+1=-12+1$$

Tāpiri -12 ki te 1.

$$x^{2}-2x+1=-11$$

Tauwehea x^{2}-2x+1. 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}.

$$\left(x-1\right)^{2}=-11$$

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

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{-11}$$

Whakarūnātia.

$$x-1=\sqrt{11}i x-1=-\sqrt{11}i$$

Me tāpiri 1 ki ngā taha e rua o te whārite.

$$x=1+\sqrt{11}i x=-\sqrt{11}i+1$$