Whakaoti i te x^2+18x+79=0

$$x ^ { 2 } + 18 x + 79 = 0$$

Whakaoti mō x (complex solution)

$$x=\sqrt{2}-9\approx -7.585786438$$

$$x=-\left(\sqrt{2}+9\right)\approx -10.414213562$$

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.

$$x^{2}+18x+79=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, 18 mō b, me 79 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-18±\sqrt{18^{2}-4\times 79}}{2}$$

Pūrua 18.

$$x=\frac{-18±\sqrt{324-4\times 79}}{2}$$

Whakareatia -4 ki te 79.

$$x=\frac{-18±\sqrt{324-316}}{2}$$

Tāpiri 324 ki te -316.

$$x=\frac{-18±\sqrt{8}}{2}$$

Tuhia te pūtakerua o te 8.

$$x=\frac{-18±2\sqrt{2}}{2}$$

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

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

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

$$x=\sqrt{2}-9$$

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

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

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

$$x=-\sqrt{2}-9$$

Kua oti te whārite te whakatau.

$$x=\sqrt{2}-9 x=-\sqrt{2}-9$$

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.

$$x^{2}+18x+79=0$$

Me tango 79 mai i ngā taha e rua o te whārite.

$$x^{2}+18x+79-79=-79$$

Mā te tango i te 79 i a ia ake anō ka toe ko te 0.

$$x^{2}+18x=-79$$

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.

$$x^{2}+18x+9^{2}=-79+9^{2}$$

Pūrua 9.

$$x^{2}+18x+81=-79+81$$

Tāpiri -79 ki te 81.

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

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}.

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

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

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

Whakarūnātia.

$$x+9=\sqrt{2} x+9=-\sqrt{2}$$

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

$$x=\sqrt{2}-9 x=-\sqrt{2}-9$$

Whakaoti mō x

$$x=\sqrt{2}-9\approx -7.585786438$$

$$x=-\sqrt{2}-9\approx -10.414213562$$

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.

$$x^{2}+18x+79=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, 18 mō b, me 79 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-18±\sqrt{18^{2}-4\times 79}}{2}$$

Pūrua 18.

$$x=\frac{-18±\sqrt{324-4\times 79}}{2}$$

Whakareatia -4 ki te 79.

$$x=\frac{-18±\sqrt{324-316}}{2}$$

Tāpiri 324 ki te -316.

$$x=\frac{-18±\sqrt{8}}{2}$$

Tuhia te pūtakerua o te 8.

$$x=\frac{-18±2\sqrt{2}}{2}$$

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

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

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

$$x=\sqrt{2}-9$$

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

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

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

$$x=-\sqrt{2}-9$$

Kua oti te whārite te whakatau.

$$x=\sqrt{2}-9 x=-\sqrt{2}-9$$

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.

$$x^{2}+18x+79=0$$

Me tango 79 mai i ngā taha e rua o te whārite.

$$x^{2}+18x+79-79=-79$$

Mā te tango i te 79 i a ia ake anō ka toe ko te 0.

$$x^{2}+18x=-79$$

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.

$$x^{2}+18x+9^{2}=-79+9^{2}$$

Pūrua 9.

$$x^{2}+18x+81=-79+81$$

Tāpiri -79 ki te 81.

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

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}.

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

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

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

Whakarūnātia.

$$x+9=\sqrt{2} x+9=-\sqrt{2}$$

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

$$x=\sqrt{2}-9 x=-\sqrt{2}-9$$