Whakaoti i te -3x^2-24x-51=0

$$- 3 x ^ { 2 } - 24 x - 51 = 0$$

Whakaoti mō x (complex solution)

$$x=-4+i$$

$$x=-4-i$$

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.

$$-3x^{2}-24x-51=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -3 mō a, -24 mō b, me -51 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-24\right)±\sqrt{\left(-24\right)^{2}-4\left(-3\right)\left(-51\right)}}{2\left(-3\right)}$$

Pūrua -24.

$$x=\frac{-\left(-24\right)±\sqrt{576-4\left(-3\right)\left(-51\right)}}{2\left(-3\right)}$$

Whakareatia -4 ki te -3.

$$x=\frac{-\left(-24\right)±\sqrt{576+12\left(-51\right)}}{2\left(-3\right)}$$

Whakareatia 12 ki te -51.

$$x=\frac{-\left(-24\right)±\sqrt{576-612}}{2\left(-3\right)}$$

Tāpiri 576 ki te -612.

$$x=\frac{-\left(-24\right)±\sqrt{-36}}{2\left(-3\right)}$$

Tuhia te pūtakerua o te -36.

$$x=\frac{-\left(-24\right)±6i}{2\left(-3\right)}$$

Ko te tauaro o -24 ko 24.

$$x=\frac{24±6i}{2\left(-3\right)}$$

Whakareatia 2 ki te -3.

$$x=\frac{24±6i}{-6}$$

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

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

Whakawehe 24+6i ki te -6.

$$x=-4-i$$

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

$$x=\frac{24-6i}{-6}$$

Whakawehe 24-6i ki te -6.

$$x=-4+i$$

Kua oti te whārite te whakatau.

$$x=-4-i x=-4+i$$

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.

$$-3x^{2}-24x-51=0$$

Me tāpiri 51 ki ngā taha e rua o te whārite.

$$-3x^{2}-24x-51-\left(-51\right)=-\left(-51\right)$$

Mā te tango i te -51 i a ia ake anō ka toe ko te 0.

$$-3x^{2}-24x=-\left(-51\right)$$

Tango -51 mai i 0.

$$-3x^{2}-24x=51$$

Whakawehea ngā taha e rua ki te -3.

$$\frac{-3x^{2}-24x}{-3}=\frac{51}{-3}$$

Mā te whakawehe ki te -3 ka wetekia te whakareanga ki te -3.

$$x^{2}+\left(-\frac{24}{-3}\right)x=\frac{51}{-3}$$

Whakawehe -24 ki te -3.

$$x^{2}+8x=\frac{51}{-3}$$

Whakawehe 51 ki te -3.

$$x^{2}+8x=-17$$

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

$$x^{2}+8x+4^{2}=-17+4^{2}$$

Pūrua 4.

$$x^{2}+8x+16=-17+16$$

Tāpiri -17 ki te 16.

$$x^{2}+8x+16=-1$$

Tauwehea x^{2}+8x+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}.

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

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

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

Whakarūnātia.

$$x+4=i x+4=-i$$

Me tango 4 mai i ngā taha e rua o te whārite.

$$x=-4+i x=-4-i$$