Whakaoti i te 9x^2-24x+21=0

$$9 { x }^{ 2 } -24x+21=0$$

Whakaoti mō x (complex solution)

$$x=\frac{4+\sqrt{5}i}{3}\approx 1.333333333+0.745355992i$$

$$x=\frac{-\sqrt{5}i+4}{3}\approx 1.333333333-0.745355992i$$

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.

$$9x^{2}-24x+21=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 9 mō a, -24 mō b, me 21 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-24\right)±\sqrt{\left(-24\right)^{2}-4\times 9\times 21}}{2\times 9}$$

Pūrua -24.

$$x=\frac{-\left(-24\right)±\sqrt{576-4\times 9\times 21}}{2\times 9}$$

Whakareatia -4 ki te 9.

$$x=\frac{-\left(-24\right)±\sqrt{576-36\times 21}}{2\times 9}$$

Whakareatia -36 ki te 21.

$$x=\frac{-\left(-24\right)±\sqrt{576-756}}{2\times 9}$$

Tāpiri 576 ki te -756.

$$x=\frac{-\left(-24\right)±\sqrt{-180}}{2\times 9}$$

Tuhia te pūtakerua o te -180.

$$x=\frac{-\left(-24\right)±6\sqrt{5}i}{2\times 9}$$

Ko te tauaro o -24 ko 24.

$$x=\frac{24±6\sqrt{5}i}{2\times 9}$$

Whakareatia 2 ki te 9.

$$x=\frac{24±6\sqrt{5}i}{18}$$

Nā, me whakaoti te whārite x=\frac{24±6\sqrt{5}i}{18} ina he tāpiri te ±. Tāpiri 24 ki te 6i\sqrt{5}.

$$x=\frac{24+6\sqrt{5}i}{18}$$

Whakawehe 24+6i\sqrt{5} ki te 18.

$$x=\frac{4+\sqrt{5}i}{3}$$

Nā, me whakaoti te whārite x=\frac{24±6\sqrt{5}i}{18} ina he tango te ±. Tango 6i\sqrt{5} mai i 24.

$$x=\frac{-6\sqrt{5}i+24}{18}$$

Whakawehe 24-6i\sqrt{5} ki te 18.

$$x=\frac{-\sqrt{5}i+4}{3}$$

Kua oti te whārite te whakatau.

$$x=\frac{4+\sqrt{5}i}{3} x=\frac{-\sqrt{5}i+4}{3}$$

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.

$$9x^{2}-24x+21=0$$

Me tango 21 mai i ngā taha e rua o te whārite.

$$9x^{2}-24x+21-21=-21$$

Mā te tango i te 21 i a ia ake anō ka toe ko te 0.

$$9x^{2}-24x=-21$$

Whakawehea ngā taha e rua ki te 9.

$$\frac{9x^{2}-24x}{9}=-\frac{21}{9}$$

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

$$x^{2}+\left(-\frac{24}{9}\right)x=-\frac{21}{9}$$

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

$$x^{2}-\frac{8}{3}x=-\frac{21}{9}$$

Whakahekea te hautanga \frac{-21}{9} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 3.

$$x^{2}-\frac{8}{3}x=-\frac{7}{3}$$

Whakawehea te -\frac{8}{3}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{4}{3}. Nā, tāpiria te pūrua o te -\frac{4}{3} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{8}{3}x+\left(-\frac{4}{3}\right)^{2}=-\frac{7}{3}+\left(-\frac{4}{3}\right)^{2}$$

Pūruatia -\frac{4}{3} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{8}{3}x+\frac{16}{9}=-\frac{7}{3}+\frac{16}{9}$$

Tāpiri -\frac{7}{3} ki te \frac{16}{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.

$$x^{2}-\frac{8}{3}x+\frac{16}{9}=-\frac{5}{9}$$

Tauwehea x^{2}-\frac{8}{3}x+\frac{16}{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}.

$$\left(x-\frac{4}{3}\right)^{2}=-\frac{5}{9}$$

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

$$\sqrt{\left(x-\frac{4}{3}\right)^{2}}=\sqrt{-\frac{5}{9}}$$

Whakarūnātia.

$$x-\frac{4}{3}=\frac{\sqrt{5}i}{3} x-\frac{4}{3}=-\frac{\sqrt{5}i}{3}$$

Me tāpiri \frac{4}{3} ki ngā taha e rua o te whārite.

$$x=\frac{4+\sqrt{5}i}{3} x=\frac{-\sqrt{5}i+4}{3}$$