Whakaoti i te x^2+18x+12=0

$$x ^ { 2 } + 18 x + 12 = 0$$

Whakaoti mō x (complex solution)

$$x=\sqrt{69}-9\approx -0.693376137$$

$$x=-\left(\sqrt{69}+9\right)\approx -17.306623863$$

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.

$$x^{2}+18x+12=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, 18 mō b, me 12 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-18±\sqrt{18^{2}-4\times 12}}{2}$$

Pūrua 18.

$$x=\frac{-18±\sqrt{324-4\times 12}}{2}$$

Whakareatia -4 ki te 12.

$$x=\frac{-18±\sqrt{324-48}}{2}$$

Tāpiri 324 ki te -48.

$$x=\frac{-18±\sqrt{276}}{2}$$

Tuhia te pūtakerua o te 276.

$$x=\frac{-18±2\sqrt{69}}{2}$$

Nā, me whakaoti te whārite x=\frac{-18±2\sqrt{69}}{2} ina he tāpiri te ±. Tāpiri -18 ki te 2\sqrt{69}.

$$x=\frac{2\sqrt{69}-18}{2}$$

Whakawehe -18+2\sqrt{69} ki te 2.

$$x=\sqrt{69}-9$$

Nā, me whakaoti te whārite x=\frac{-18±2\sqrt{69}}{2} ina he tango te ±. Tango 2\sqrt{69} mai i -18.

$$x=\frac{-2\sqrt{69}-18}{2}$$

Whakawehe -18-2\sqrt{69} ki te 2.

$$x=-\sqrt{69}-9$$

Kua oti te whārite te whakatau.

$$x=\sqrt{69}-9 x=-\sqrt{69}-9$$

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.

$$x^{2}+18x+12=0$$

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

$$x^{2}+18x+12-12=-12$$

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

$$x^{2}+18x=-12$$

Whakawehea te 18, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 9. Nā, tāpiria te pūrua o te 9 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+18x+9^{2}=-12+9^{2}$$

Pūrua 9.

$$x^{2}+18x+81=-12+81$$

Tāpiri -12 ki te 81.

$$x^{2}+18x+81=69$$

Tauwehea x^{2}+18x+81. 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}.

$$\left(x+9\right)^{2}=69$$

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

$$\sqrt{\left(x+9\right)^{2}}=\sqrt{69}$$

Whakarūnātia.

$$x+9=\sqrt{69} x+9=-\sqrt{69}$$

Me tango 9 mai i ngā taha e rua o te whārite.

$$x=\sqrt{69}-9 x=-\sqrt{69}-9$$

Whakaoti mō x

$$x=\sqrt{69}-9\approx -0.693376137$$

$$x=-\sqrt{69}-9\approx -17.306623863$$

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.

$$x^{2}+18x+12=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, 18 mō b, me 12 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-18±\sqrt{18^{2}-4\times 12}}{2}$$

Pūrua 18.

$$x=\frac{-18±\sqrt{324-4\times 12}}{2}$$

Whakareatia -4 ki te 12.

$$x=\frac{-18±\sqrt{324-48}}{2}$$

Tāpiri 324 ki te -48.

$$x=\frac{-18±\sqrt{276}}{2}$$

Tuhia te pūtakerua o te 276.

$$x=\frac{-18±2\sqrt{69}}{2}$$

Nā, me whakaoti te whārite x=\frac{-18±2\sqrt{69}}{2} ina he tāpiri te ±. Tāpiri -18 ki te 2\sqrt{69}.

$$x=\frac{2\sqrt{69}-18}{2}$$

Whakawehe -18+2\sqrt{69} ki te 2.

$$x=\sqrt{69}-9$$

Nā, me whakaoti te whārite x=\frac{-18±2\sqrt{69}}{2} ina he tango te ±. Tango 2\sqrt{69} mai i -18.

$$x=\frac{-2\sqrt{69}-18}{2}$$

Whakawehe -18-2\sqrt{69} ki te 2.

$$x=-\sqrt{69}-9$$

Kua oti te whārite te whakatau.

$$x=\sqrt{69}-9 x=-\sqrt{69}-9$$

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.

$$x^{2}+18x+12=0$$

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

$$x^{2}+18x+12-12=-12$$

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

$$x^{2}+18x=-12$$

Whakawehea te 18, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 9. Nā, tāpiria te pūrua o te 9 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+18x+9^{2}=-12+9^{2}$$

Pūrua 9.

$$x^{2}+18x+81=-12+81$$

Tāpiri -12 ki te 81.

$$x^{2}+18x+81=69$$

Tauwehea x^{2}+18x+81. 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}.

$$\left(x+9\right)^{2}=69$$

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

$$\sqrt{\left(x+9\right)^{2}}=\sqrt{69}$$

Whakarūnātia.

$$x+9=\sqrt{69} x+9=-\sqrt{69}$$

Me tango 9 mai i ngā taha e rua o te whārite.

$$x=\sqrt{69}-9 x=-\sqrt{69}-9$$