Whakaoti i te -6x^2+12x-486=0

$$- 6 x ^ { 2 } + 12 x - 486 = 0$$

Whakaoti mō x (complex solution)

$$x=1+4\sqrt{5}i\approx 1+8.94427191i$$

$$x=-4\sqrt{5}i+1\approx 1-8.94427191i$$

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.

$$-6x^{2}+12x-486=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -6 mō a, 12 mō b, me -486 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-12±\sqrt{12^{2}-4\left(-6\right)\left(-486\right)}}{2\left(-6\right)}$$

Pūrua 12.

$$x=\frac{-12±\sqrt{144-4\left(-6\right)\left(-486\right)}}{2\left(-6\right)}$$

Whakareatia -4 ki te -6.

$$x=\frac{-12±\sqrt{144+24\left(-486\right)}}{2\left(-6\right)}$$

Whakareatia 24 ki te -486.

$$x=\frac{-12±\sqrt{144-11664}}{2\left(-6\right)}$$

Tāpiri 144 ki te -11664.

$$x=\frac{-12±\sqrt{-11520}}{2\left(-6\right)}$$

Tuhia te pūtakerua o te -11520.

$$x=\frac{-12±48\sqrt{5}i}{2\left(-6\right)}$$

Whakareatia 2 ki te -6.

$$x=\frac{-12±48\sqrt{5}i}{-12}$$

Nā, me whakaoti te whārite x=\frac{-12±48\sqrt{5}i}{-12} ina he tāpiri te ±. Tāpiri -12 ki te 48i\sqrt{5}.

$$x=\frac{-12+48\sqrt{5}i}{-12}$$

Whakawehe -12+48i\sqrt{5} ki te -12.

$$x=-4\sqrt{5}i+1$$

Nā, me whakaoti te whārite x=\frac{-12±48\sqrt{5}i}{-12} ina he tango te ±. Tango 48i\sqrt{5} mai i -12.

$$x=\frac{-48\sqrt{5}i-12}{-12}$$

Whakawehe -12-48i\sqrt{5} ki te -12.

$$x=1+4\sqrt{5}i$$

Kua oti te whārite te whakatau.

$$x=-4\sqrt{5}i+1 x=1+4\sqrt{5}i$$

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.

$$-6x^{2}+12x-486=0$$

Me tāpiri 486 ki ngā taha e rua o te whārite.

$$-6x^{2}+12x-486-\left(-486\right)=-\left(-486\right)$$

Mā te tango i te -486 i a ia ake anō ka toe ko te 0.

$$-6x^{2}+12x=-\left(-486\right)$$

Tango -486 mai i 0.

$$-6x^{2}+12x=486$$

Whakawehea ngā taha e rua ki te -6.

$$\frac{-6x^{2}+12x}{-6}=\frac{486}{-6}$$

Mā te whakawehe ki te -6 ka wetekia te whakareanga ki te -6.

$$x^{2}+\frac{12}{-6}x=\frac{486}{-6}$$

Whakawehe 12 ki te -6.

$$x^{2}-2x=\frac{486}{-6}$$

Whakawehe 486 ki te -6.

$$x^{2}-2x=-81$$

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.

$$x^{2}-2x+1=-81+1$$

Tāpiri -81 ki te 1.

$$x^{2}-2x+1=-80$$

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}.

$$\left(x-1\right)^{2}=-80$$

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

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{-80}$$

Whakarūnātia.

$$x-1=4\sqrt{5}i x-1=-4\sqrt{5}i$$

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

$$x=1+4\sqrt{5}i x=-4\sqrt{5}i+1$$