Whakaoti i te -3x^2+32x-160=0

$$- 3 x ^ { 2 } + 32 x - 160 = 0$$

Whakaoti mō x (complex solution)

$$x=\frac{-4\sqrt{14}i+16}{3}\approx 5.333333333-4.988876516i$$

$$x=\frac{16+4\sqrt{14}i}{3}\approx 5.333333333+4.988876516i$$

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.

$$-3x^{2}+32x-160=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -3 mō a, 32 mō b, me -160 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-32±\sqrt{32^{2}-4\left(-3\right)\left(-160\right)}}{2\left(-3\right)}$$

Pūrua 32.

$$x=\frac{-32±\sqrt{1024-4\left(-3\right)\left(-160\right)}}{2\left(-3\right)}$$

Whakareatia -4 ki te -3.

$$x=\frac{-32±\sqrt{1024+12\left(-160\right)}}{2\left(-3\right)}$$

Whakareatia 12 ki te -160.

$$x=\frac{-32±\sqrt{1024-1920}}{2\left(-3\right)}$$

Tāpiri 1024 ki te -1920.

$$x=\frac{-32±\sqrt{-896}}{2\left(-3\right)}$$

Tuhia te pūtakerua o te -896.

$$x=\frac{-32±8\sqrt{14}i}{2\left(-3\right)}$$

Whakareatia 2 ki te -3.

$$x=\frac{-32±8\sqrt{14}i}{-6}$$

Nā, me whakaoti te whārite x=\frac{-32±8\sqrt{14}i}{-6} ina he tāpiri te ±. Tāpiri -32 ki te 8i\sqrt{14}.

$$x=\frac{-32+8\sqrt{14}i}{-6}$$

Whakawehe -32+8i\sqrt{14} ki te -6.

$$x=\frac{-4\sqrt{14}i+16}{3}$$

Nā, me whakaoti te whārite x=\frac{-32±8\sqrt{14}i}{-6} ina he tango te ±. Tango 8i\sqrt{14} mai i -32.

$$x=\frac{-8\sqrt{14}i-32}{-6}$$

Whakawehe -32-8i\sqrt{14} ki te -6.

$$x=\frac{16+4\sqrt{14}i}{3}$$

Kua oti te whārite te whakatau.

$$x=\frac{-4\sqrt{14}i+16}{3} x=\frac{16+4\sqrt{14}i}{3}$$

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.

$$-3x^{2}+32x-160=0$$

Me tāpiri 160 ki ngā taha e rua o te whārite.

$$-3x^{2}+32x-160-\left(-160\right)=-\left(-160\right)$$

Mā te tango i te -160 i a ia ake anō ka toe ko te 0.

$$-3x^{2}+32x=-\left(-160\right)$$

Tango -160 mai i 0.

$$-3x^{2}+32x=160$$

Whakawehea ngā taha e rua ki te -3.

$$\frac{-3x^{2}+32x}{-3}=\frac{160}{-3}$$

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

$$x^{2}+\frac{32}{-3}x=\frac{160}{-3}$$

Whakawehe 32 ki te -3.

$$x^{2}-\frac{32}{3}x=\frac{160}{-3}$$

Whakawehe 160 ki te -3.

$$x^{2}-\frac{32}{3}x=-\frac{160}{3}$$

Whakawehea te -\frac{32}{3}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{16}{3}. Nā, tāpiria te pūrua o te -\frac{16}{3} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{32}{3}x+\left(-\frac{16}{3}\right)^{2}=-\frac{160}{3}+\left(-\frac{16}{3}\right)^{2}$$

Pūruatia -\frac{16}{3} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{32}{3}x+\frac{256}{9}=-\frac{160}{3}+\frac{256}{9}$$

Tāpiri -\frac{160}{3} ki te \frac{256}{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.

$$x^{2}-\frac{32}{3}x+\frac{256}{9}=-\frac{224}{9}$$

Tauwehea x^{2}-\frac{32}{3}x+\frac{256}{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}.

$$\left(x-\frac{16}{3}\right)^{2}=-\frac{224}{9}$$

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

$$\sqrt{\left(x-\frac{16}{3}\right)^{2}}=\sqrt{-\frac{224}{9}}$$

Whakarūnātia.

$$x-\frac{16}{3}=\frac{4\sqrt{14}i}{3} x-\frac{16}{3}=-\frac{4\sqrt{14}i}{3}$$

Me tāpiri \frac{16}{3} ki ngā taha e rua o te whārite.

$$x=\frac{16+4\sqrt{14}i}{3} x=\frac{-4\sqrt{14}i+16}{3}$$