Whakaoti i te 3x^2-6x+1=0

$$3 { x }^{ 2 } -6x+1=0$$

Whakaoti mō x

$$x=\frac{\sqrt{6}}{3}+1\approx 1.816496581$$

$$x=-\frac{\sqrt{6}}{3}+1\approx 0.183503419$$

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.

$$3x^{2}-6x+1=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 3 mō a, -6 mō b, me 1 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

Pūrua -6.

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

Whakareatia -4 ki te 3.

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

Tāpiri 36 ki te -12.

$$x=\frac{-\left(-6\right)±\sqrt{24}}{2\times 3}$$

Tuhia te pūtakerua o te 24.

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

Ko te tauaro o -6 ko 6.

$$x=\frac{6±2\sqrt{6}}{2\times 3}$$

Whakareatia 2 ki te 3.

$$x=\frac{6±2\sqrt{6}}{6}$$

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

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

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

$$x=\frac{\sqrt{6}}{3}+1$$

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

$$x=\frac{6-2\sqrt{6}}{6}$$

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

$$x=-\frac{\sqrt{6}}{3}+1$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{6}}{3}+1 x=-\frac{\sqrt{6}}{3}+1$$

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.

$$3x^{2}-6x+1=0$$

Me tango 1 mai i ngā taha e rua o te whārite.

$$3x^{2}-6x+1-1=-1$$

Mā te tango i te 1 i a ia ake anō ka toe ko te 0.

$$3x^{2}-6x=-1$$

Whakawehea ngā taha e rua ki te 3.

$$\frac{3x^{2}-6x}{3}=-\frac{1}{3}$$

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

$$x^{2}+\left(-\frac{6}{3}\right)x=-\frac{1}{3}$$

Whakawehe -6 ki te 3.

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

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.

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

Tāpiri -\frac{1}{3} ki te 1.

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

Tauwehea te x^{2}-2x+1. Ko te tikanga, ina ko x^{2}+bx+c he pūrua tika, ka taea te tauwehe i ngā wā katoa hei \left(x+\frac{b}{2}\right)^{2}.

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

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

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

Whakarūnātia.

$$x-1=\frac{\sqrt{6}}{3} x-1=-\frac{\sqrt{6}}{3}$$

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

$$x=\frac{\sqrt{6}}{3}+1 x=-\frac{\sqrt{6}}{3}+1$$