Whakaoti i te x^2-12x-9=0

$$x ^ { 2 } - 12 x - 9 = 0$$

Whakaoti mō x

$$x=3\sqrt{5}+6\approx 12.708203932$$

$$x=6-3\sqrt{5}\approx -0.708203932$$

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.

$$x^{2}-12x-9=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -12 mō b, me -9 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\left(-9\right)}}{2}$$

Pūrua -12.

$$x=\frac{-\left(-12\right)±\sqrt{144-4\left(-9\right)}}{2}$$

Whakareatia -4 ki te -9.

$$x=\frac{-\left(-12\right)±\sqrt{144+36}}{2}$$

Tāpiri 144 ki te 36.

$$x=\frac{-\left(-12\right)±\sqrt{180}}{2}$$

Tuhia te pūtakerua o te 180.

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

Ko te tauaro o -12 ko 12.

$$x=\frac{12±6\sqrt{5}}{2}$$

Nā, me whakaoti te whārite x=\frac{12±6\sqrt{5}}{2} ina he tāpiri te ±. Tāpiri 12 ki te 6\sqrt{5}.

$$x=\frac{6\sqrt{5}+12}{2}$$

Whakawehe 12+6\sqrt{5} ki te 2.

$$x=3\sqrt{5}+6$$

Nā, me whakaoti te whārite x=\frac{12±6\sqrt{5}}{2} ina he tango te ±. Tango 6\sqrt{5} mai i 12.

$$x=\frac{12-6\sqrt{5}}{2}$$

Whakawehe 12-6\sqrt{5} ki te 2.

$$x=6-3\sqrt{5}$$

Kua oti te whārite te whakatau.

$$x=3\sqrt{5}+6 x=6-3\sqrt{5}$$

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.

$$x^{2}-12x-9=0$$

Me tāpiri 9 ki ngā taha e rua o te whārite.

$$x^{2}-12x-9-\left(-9\right)=-\left(-9\right)$$

Mā te tango i te -9 i a ia ake anō ka toe ko te 0.

$$x^{2}-12x=-\left(-9\right)$$

Tango -9 mai i 0.

$$x^{2}-12x=9$$

Whakawehea te -12, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -6. Nā, tāpiria te pūrua o te -6 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-12x+\left(-6\right)^{2}=9+\left(-6\right)^{2}$$

Pūrua -6.

$$x^{2}-12x+36=9+36$$

Tāpiri 9 ki te 36.

$$x^{2}-12x+36=45$$

Tauwehea x^{2}-12x+36. 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}.

$$\left(x-6\right)^{2}=45$$

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

$$\sqrt{\left(x-6\right)^{2}}=\sqrt{45}$$

Whakarūnātia.

$$x-6=3\sqrt{5} x-6=-3\sqrt{5}$$

Me tāpiri 6 ki ngā taha e rua o te whārite.

$$x=3\sqrt{5}+6 x=6-3\sqrt{5}$$