Whakaoti i te -x^2-8x+12=0

$$- x ^ { 2 } - 8 x + 12 = 0$$

Whakaoti mō x

$$x=2\sqrt{7}-4\approx 1.291502622$$

$$x=-2\sqrt{7}-4\approx -9.291502622$$

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.

$$-x^{2}-8x+12=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -1 mō a, -8 mō b, me 12 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\left(-1\right)\times 12}}{2\left(-1\right)}$$

Pūrua -8.

$$x=\frac{-\left(-8\right)±\sqrt{64-4\left(-1\right)\times 12}}{2\left(-1\right)}$$

Whakareatia -4 ki te -1.

$$x=\frac{-\left(-8\right)±\sqrt{64+4\times 12}}{2\left(-1\right)}$$

Whakareatia 4 ki te 12.

$$x=\frac{-\left(-8\right)±\sqrt{64+48}}{2\left(-1\right)}$$

Tāpiri 64 ki te 48.

$$x=\frac{-\left(-8\right)±\sqrt{112}}{2\left(-1\right)}$$

Tuhia te pūtakerua o te 112.

$$x=\frac{-\left(-8\right)±4\sqrt{7}}{2\left(-1\right)}$$

Ko te tauaro o -8 ko 8.

$$x=\frac{8±4\sqrt{7}}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$x=\frac{8±4\sqrt{7}}{-2}$$

Nā, me whakaoti te whārite x=\frac{8±4\sqrt{7}}{-2} ina he tāpiri te ±. Tāpiri 8 ki te 4\sqrt{7}.

$$x=\frac{4\sqrt{7}+8}{-2}$$

Whakawehe 8+4\sqrt{7} ki te -2.

$$x=-2\sqrt{7}-4$$

Nā, me whakaoti te whārite x=\frac{8±4\sqrt{7}}{-2} ina he tango te ±. Tango 4\sqrt{7} mai i 8.

$$x=\frac{8-4\sqrt{7}}{-2}$$

Whakawehe 8-4\sqrt{7} ki te -2.

$$x=2\sqrt{7}-4$$

Kua oti te whārite te whakatau.

$$x=-2\sqrt{7}-4 x=2\sqrt{7}-4$$

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.

$$-x^{2}-8x+12=0$$

Me tango 12 mai i ngā taha e rua o te whārite.

$$-x^{2}-8x+12-12=-12$$

Mā te tango i te 12 i a ia ake anō ka toe ko te 0.

$$-x^{2}-8x=-12$$

Whakawehea ngā taha e rua ki te -1.

$$\frac{-x^{2}-8x}{-1}=-\frac{12}{-1}$$

Mā te whakawehe ki te -1 ka wetekia te whakareanga ki te -1.

$$x^{2}+\left(-\frac{8}{-1}\right)x=-\frac{12}{-1}$$

Whakawehe -8 ki te -1.

$$x^{2}+8x=-\frac{12}{-1}$$

Whakawehe -12 ki te -1.

$$x^{2}+8x=12$$

Whakawehea te 8, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 4. Nā, tāpiria te pūrua o te 4 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+8x+4^{2}=12+4^{2}$$

Pūrua 4.

$$x^{2}+8x+16=12+16$$

Tāpiri 12 ki te 16.

$$x^{2}+8x+16=28$$

Tauwehea te x^{2}+8x+16. 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}.

$$\left(x+4\right)^{2}=28$$

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

$$\sqrt{\left(x+4\right)^{2}}=\sqrt{28}$$

Whakarūnātia.

$$x+4=2\sqrt{7} x+4=-2\sqrt{7}$$

Me tango 4 mai i ngā taha e rua o te whārite.

$$x=2\sqrt{7}-4 x=-2\sqrt{7}-4$$