Whakaoti i te 3x^2*16-24x+24=0

$$3 { x }^{ 2 } \times 16-24x+24=0$$

Whakaoti mō x (complex solution)

$$x=\frac{1+\sqrt{7}i}{4}\approx 0.25+0.661437828i$$

$$x=\frac{-\sqrt{7}i+1}{4}\approx 0.25-0.661437828i$$

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.

$$48x^{2}-24x+24=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 48 mō a, -24 mō b, me 24 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-24\right)±\sqrt{\left(-24\right)^{2}-4\times 48\times 24}}{2\times 48}$$

Pūrua -24.

$$x=\frac{-\left(-24\right)±\sqrt{576-4\times 48\times 24}}{2\times 48}$$

Whakareatia -4 ki te 48.

$$x=\frac{-\left(-24\right)±\sqrt{576-192\times 24}}{2\times 48}$$

Whakareatia -192 ki te 24.

$$x=\frac{-\left(-24\right)±\sqrt{576-4608}}{2\times 48}$$

Tāpiri 576 ki te -4608.

$$x=\frac{-\left(-24\right)±\sqrt{-4032}}{2\times 48}$$

Tuhia te pūtakerua o te -4032.

$$x=\frac{-\left(-24\right)±24\sqrt{7}i}{2\times 48}$$

Ko te tauaro o -24 ko 24.

$$x=\frac{24±24\sqrt{7}i}{2\times 48}$$

Whakareatia 2 ki te 48.

$$x=\frac{24±24\sqrt{7}i}{96}$$

Nā, me whakaoti te whārite x=\frac{24±24\sqrt{7}i}{96} ina he tāpiri te ±. Tāpiri 24 ki te 24i\sqrt{7}.

$$x=\frac{24+24\sqrt{7}i}{96}$$

Whakawehe 24+24i\sqrt{7} ki te 96.

$$x=\frac{1+\sqrt{7}i}{4}$$

Nā, me whakaoti te whārite x=\frac{24±24\sqrt{7}i}{96} ina he tango te ±. Tango 24i\sqrt{7} mai i 24.

$$x=\frac{-24\sqrt{7}i+24}{96}$$

Whakawehe 24-24i\sqrt{7} ki te 96.

$$x=\frac{-\sqrt{7}i+1}{4}$$

Kua oti te whārite te whakatau.

$$x=\frac{1+\sqrt{7}i}{4} x=\frac{-\sqrt{7}i+1}{4}$$

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.

$$48x^{2}-24x+24=0$$

Me tango 24 mai i ngā taha e rua o te whārite.

$$48x^{2}-24x+24-24=-24$$

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

$$48x^{2}-24x=-24$$

Whakawehea ngā taha e rua ki te 48.

$$\frac{48x^{2}-24x}{48}=-\frac{24}{48}$$

Mā te whakawehe ki te 48 ka wetekia te whakareanga ki te 48.

$$x^{2}+\left(-\frac{24}{48}\right)x=-\frac{24}{48}$$

Whakahekea te hautanga \frac{-24}{48} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 24.

$$x^{2}-\frac{1}{2}x=-\frac{24}{48}$$

Whakahekea te hautanga \frac{-24}{48} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 24.

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

Whakawehea te -\frac{1}{2}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{1}{4}. Nā, tāpiria te pūrua o te -\frac{1}{4} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

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

Pūruatia -\frac{1}{4} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{1}{2}x+\frac{1}{16}=-\frac{1}{2}+\frac{1}{16}$$

Tāpiri -\frac{1}{2} ki te \frac{1}{16} 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.

$$x^{2}-\frac{1}{2}x+\frac{1}{16}=-\frac{7}{16}$$

Tauwehea x^{2}-\frac{1}{2}x+\frac{1}{16}. 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}.

$$\left(x-\frac{1}{4}\right)^{2}=-\frac{7}{16}$$

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

$$\sqrt{\left(x-\frac{1}{4}\right)^{2}}=\sqrt{-\frac{7}{16}}$$

Whakarūnātia.

$$x-\frac{1}{4}=\frac{\sqrt{7}i}{4} x-\frac{1}{4}=-\frac{\sqrt{7}i}{4}$$

Me tāpiri \frac{1}{4} ki ngā taha e rua o te whārite.

$$x=\frac{1+\sqrt{7}i}{4} x=\frac{-\sqrt{7}i+1}{4}$$