Whakaoti i te 9x^2+12x-24=0

$$9 x ^ { 2 } + 12 x - 24 = 0$$

Whakaoti mō x

$$x = \frac{2 \sqrt{7} - 2}{3} \approx 1.097167541$$

$$x=\frac{-2\sqrt{7}-2}{3}\approx -2.430500874$$

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.

$$9x^{2}+12x-24=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 9 mō a, 12 mō b, me -24 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-12±\sqrt{12^{2}-4\times 9\left(-24\right)}}{2\times 9}$$

Pūrua 12.

$$x=\frac{-12±\sqrt{144-4\times 9\left(-24\right)}}{2\times 9}$$

Whakareatia -4 ki te 9.

$$x=\frac{-12±\sqrt{144-36\left(-24\right)}}{2\times 9}$$

Whakareatia -36 ki te -24.

$$x=\frac{-12±\sqrt{144+864}}{2\times 9}$$

Tāpiri 144 ki te 864.

$$x=\frac{-12±\sqrt{1008}}{2\times 9}$$

Tuhia te pūtakerua o te 1008.

$$x=\frac{-12±12\sqrt{7}}{2\times 9}$$

Whakareatia 2 ki te 9.

$$x=\frac{-12±12\sqrt{7}}{18}$$

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

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

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

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

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

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

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

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

Kua oti te whārite te whakatau.

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

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.

$$9x^{2}+12x-24=0$$

Me tāpiri 24 ki ngā taha e rua o te whārite.

$$9x^{2}+12x-24-\left(-24\right)=-\left(-24\right)$$

Mā te tango i te -24 i a ia ake anō ka toe ko te 0.

$$9x^{2}+12x=-\left(-24\right)$$

Tango -24 mai i 0.

$$9x^{2}+12x=24$$

Whakawehea ngā taha e rua ki te 9.

$$\frac{9x^{2}+12x}{9}=\frac{24}{9}$$

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

$$x^{2}+\frac{12}{9}x=\frac{24}{9}$$

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

$$x^{2}+\frac{4}{3}x=\frac{24}{9}$$

Whakahekea te hautanga \frac{24}{9} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 3.

$$x^{2}+\frac{4}{3}x=\frac{8}{3}$$

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.

$$x^{2}+\frac{4}{3}x+\left(\frac{2}{3}\right)^{2}=\frac{8}{3}+\left(\frac{2}{3}\right)^{2}$$

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

$$x^{2}+\frac{4}{3}x+\frac{4}{9}=\frac{8}{3}+\frac{4}{9}$$

Tāpiri \frac{8}{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.

$$x^{2}+\frac{4}{3}x+\frac{4}{9}=\frac{28}{9}$$

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}.

$$\left(x+\frac{2}{3}\right)^{2}=\frac{28}{9}$$

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

$$\sqrt{\left(x+\frac{2}{3}\right)^{2}}=\sqrt{\frac{28}{9}}$$

Whakarūnātia.

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

Me tango \frac{2}{3} mai i ngā taha e rua o te whārite.

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