Whakaoti i te 9x^2-12x-3=0

$$9 { x }^{ 2 } -12x-3=0$$

Whakaoti mō x

$$x = \frac{\sqrt{7} + 2}{3} \approx 1.54858377$$

$$x=\frac{2-\sqrt{7}}{3}\approx -0.215250437$$

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.

$$9x^{2}-12x-3=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 9 mō a, -12 mō b, me -3 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\times 9\left(-3\right)}}{2\times 9}$$

Pūrua -12.

$$x=\frac{-\left(-12\right)±\sqrt{144-4\times 9\left(-3\right)}}{2\times 9}$$

Whakareatia -4 ki te 9.

$$x=\frac{-\left(-12\right)±\sqrt{144-36\left(-3\right)}}{2\times 9}$$

Whakareatia -36 ki te -3.

$$x=\frac{-\left(-12\right)±\sqrt{144+108}}{2\times 9}$$

Tāpiri 144 ki te 108.

$$x=\frac{-\left(-12\right)±\sqrt{252}}{2\times 9}$$

Tuhia te pūtakerua o te 252.

$$x=\frac{-\left(-12\right)±6\sqrt{7}}{2\times 9}$$

Ko te tauaro o -12 ko 12.

$$x=\frac{12±6\sqrt{7}}{2\times 9}$$

Whakareatia 2 ki te 9.

$$x=\frac{12±6\sqrt{7}}{18}$$

Nā, me whakaoti te whārite x=\frac{12±6\sqrt{7}}{18} ina he tāpiri te ±. Tāpiri 12 ki te 6\sqrt{7}.

$$x=\frac{6\sqrt{7}+12}{18}$$

Whakawehe 12+6\sqrt{7} ki te 18.

$$x=\frac{\sqrt{7}+2}{3}$$

Nā, me whakaoti te whārite x=\frac{12±6\sqrt{7}}{18} ina he tango te ±. Tango 6\sqrt{7} mai i 12.

$$x=\frac{12-6\sqrt{7}}{18}$$

Whakawehe 12-6\sqrt{7} ki te 18.

$$x=\frac{2-\sqrt{7}}{3}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{7}+2}{3} x=\frac{2-\sqrt{7}}{3}$$

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.

$$9x^{2}-12x-3=0$$

Me tāpiri 3 ki ngā taha e rua o te whārite.

$$9x^{2}-12x-3-\left(-3\right)=-\left(-3\right)$$

Mā te tango i te -3 i a ia ake anō ka toe ko te 0.

$$9x^{2}-12x=-\left(-3\right)$$

Tango -3 mai i 0.

$$9x^{2}-12x=3$$

Whakawehea ngā taha e rua ki te 9.

$$\frac{9x^{2}-12x}{9}=\frac{3}{9}$$

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

$$x^{2}+\left(-\frac{12}{9}\right)x=\frac{3}{9}$$

Whakahekea te hautanga \frac{-12}{9} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 3.

$$x^{2}-\frac{4}{3}x=\frac{3}{9}$$

Whakahekea te hautanga \frac{3}{9} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 3.

$$x^{2}-\frac{4}{3}x=\frac{1}{3}$$

Whakawehea te -\frac{4}{3}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{2}{3}. Nā, tāpiria te pūrua o te -\frac{2}{3} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{4}{3}x+\left(-\frac{2}{3}\right)^{2}=\frac{1}{3}+\left(-\frac{2}{3}\right)^{2}$$

Pūruatia -\frac{2}{3} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{4}{3}x+\frac{4}{9}=\frac{1}{3}+\frac{4}{9}$$

Tāpiri \frac{1}{3} ki te \frac{4}{9} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x^{2}-\frac{4}{3}x+\frac{4}{9}=\frac{7}{9}$$

Tauwehea x^{2}-\frac{4}{3}x+\frac{4}{9}. 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}.

$$\left(x-\frac{2}{3}\right)^{2}=\frac{7}{9}$$

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

$$\sqrt{\left(x-\frac{2}{3}\right)^{2}}=\sqrt{\frac{7}{9}}$$

Whakarūnātia.

$$x-\frac{2}{3}=\frac{\sqrt{7}}{3} x-\frac{2}{3}=-\frac{\sqrt{7}}{3}$$

Me tāpiri \frac{2}{3} ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{7}+2}{3} x=\frac{2-\sqrt{7}}{3}$$