Whakaoti i te -9x^2+18x+68=0

$$- 9 x ^ { 2 } + 18 x + 68 = 0$$

Whakaoti mō x

$$x=\frac{\sqrt{77}}{3}+1\approx 3.924988129$$

$$x=-\frac{\sqrt{77}}{3}+1\approx -1.924988129$$

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.

$$-9x^{2}+18x+68=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -9 mō a, 18 mō b, me 68 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-18±\sqrt{18^{2}-4\left(-9\right)\times 68}}{2\left(-9\right)}$$

Pūrua 18.

$$x=\frac{-18±\sqrt{324-4\left(-9\right)\times 68}}{2\left(-9\right)}$$

Whakareatia -4 ki te -9.

$$x=\frac{-18±\sqrt{324+36\times 68}}{2\left(-9\right)}$$

Whakareatia 36 ki te 68.

$$x=\frac{-18±\sqrt{324+2448}}{2\left(-9\right)}$$

Tāpiri 324 ki te 2448.

$$x=\frac{-18±\sqrt{2772}}{2\left(-9\right)}$$

Tuhia te pūtakerua o te 2772.

$$x=\frac{-18±6\sqrt{77}}{2\left(-9\right)}$$

Whakareatia 2 ki te -9.

$$x=\frac{-18±6\sqrt{77}}{-18}$$

Nā, me whakaoti te whārite x=\frac{-18±6\sqrt{77}}{-18} ina he tāpiri te ±. Tāpiri -18 ki te 6\sqrt{77}.

$$x=\frac{6\sqrt{77}-18}{-18}$$

Whakawehe -18+6\sqrt{77} ki te -18.

$$x=-\frac{\sqrt{77}}{3}+1$$

Nā, me whakaoti te whārite x=\frac{-18±6\sqrt{77}}{-18} ina he tango te ±. Tango 6\sqrt{77} mai i -18.

$$x=\frac{-6\sqrt{77}-18}{-18}$$

Whakawehe -18-6\sqrt{77} ki te -18.

$$x=\frac{\sqrt{77}}{3}+1$$

Kua oti te whārite te whakatau.

$$x=-\frac{\sqrt{77}}{3}+1 x=\frac{\sqrt{77}}{3}+1$$

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.

$$-9x^{2}+18x+68=0$$

Me tango 68 mai i ngā taha e rua o te whārite.

$$-9x^{2}+18x+68-68=-68$$

Mā te tango i te 68 i a ia ake anō ka toe ko te 0.

$$-9x^{2}+18x=-68$$

Whakawehea ngā taha e rua ki te -9.

$$\frac{-9x^{2}+18x}{-9}=-\frac{68}{-9}$$

Mā te whakawehe ki te -9 ka wetekia te whakareanga ki te -9.

$$x^{2}+\frac{18}{-9}x=-\frac{68}{-9}$$

Whakawehe 18 ki te -9.

$$x^{2}-2x=-\frac{68}{-9}$$

Whakawehe -68 ki te -9.

$$x^{2}-2x=\frac{68}{9}$$

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.

$$x^{2}-2x+1=\frac{68}{9}+1$$

Tāpiri \frac{68}{9} ki te 1.

$$x^{2}-2x+1=\frac{77}{9}$$

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}.

$$\left(x-1\right)^{2}=\frac{77}{9}$$

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

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{\frac{77}{9}}$$

Whakarūnātia.

$$x-1=\frac{\sqrt{77}}{3} x-1=-\frac{\sqrt{77}}{3}$$

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

$$x=\frac{\sqrt{77}}{3}+1 x=-\frac{\sqrt{77}}{3}+1$$