Whakaoti i te 3x^2-12x+6=0

$$3 x ^ { 2 } - 12 x + 6 = 0$$

Whakaoti mō x

$$x=\sqrt{2}+2\approx 3.414213562$$

$$x=2-\sqrt{2}\approx 0.585786438$$

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.

$$3x^{2}-12x+6=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 3 mō a, -12 mō b, me 6 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\times 3\times 6}}{2\times 3}$$

Pūrua -12.

$$x=\frac{-\left(-12\right)±\sqrt{144-4\times 3\times 6}}{2\times 3}$$

Whakareatia -4 ki te 3.

$$x=\frac{-\left(-12\right)±\sqrt{144-12\times 6}}{2\times 3}$$

Whakareatia -12 ki te 6.

$$x=\frac{-\left(-12\right)±\sqrt{144-72}}{2\times 3}$$

Tāpiri 144 ki te -72.

$$x=\frac{-\left(-12\right)±\sqrt{72}}{2\times 3}$$

Tuhia te pūtakerua o te 72.

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

Ko te tauaro o -12 ko 12.

$$x=\frac{12±6\sqrt{2}}{2\times 3}$$

Whakareatia 2 ki te 3.

$$x=\frac{12±6\sqrt{2}}{6}$$

Nā, me whakaoti te whārite x=\frac{12±6\sqrt{2}}{6} ina he tāpiri te ±. Tāpiri 12 ki te 6\sqrt{2}.

$$x=\frac{6\sqrt{2}+12}{6}$$

Whakawehe 12+6\sqrt{2} ki te 6.

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

Nā, me whakaoti te whārite x=\frac{12±6\sqrt{2}}{6} ina he tango te ±. Tango 6\sqrt{2} mai i 12.

$$x=\frac{12-6\sqrt{2}}{6}$$

Whakawehe 12-6\sqrt{2} ki te 6.

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

Kua oti te whārite te whakatau.

$$x=\sqrt{2}+2 x=2-\sqrt{2}$$

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.

$$3x^{2}-12x+6=0$$

Me tango 6 mai i ngā taha e rua o te whārite.

$$3x^{2}-12x+6-6=-6$$

Mā te tango i te 6 i a ia ake anō ka toe ko te 0.

$$3x^{2}-12x=-6$$

Whakawehea ngā taha e rua ki te 3.

$$\frac{3x^{2}-12x}{3}=-\frac{6}{3}$$

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

$$x^{2}+\left(-\frac{12}{3}\right)x=-\frac{6}{3}$$

Whakawehe -12 ki te 3.

$$x^{2}-4x=-\frac{6}{3}$$

Whakawehe -6 ki te 3.

$$x^{2}-4x=-2$$

Whakawehea te -4, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -2. Nā, tāpiria te pūrua o te -2 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-4x+\left(-2\right)^{2}=-2+\left(-2\right)^{2}$$

Pūrua -2.

$$x^{2}-4x+4=-2+4$$

Tāpiri -2 ki te 4.

$$x^{2}-4x+4=2$$

Tauwehea x^{2}-4x+4. 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}.

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

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

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

Whakarūnātia.

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

Me tāpiri 2 ki ngā taha e rua o te whārite.

$$x=\sqrt{2}+2 x=2-\sqrt{2}$$