Whakaoti i te 28x-6x^2=80

$$28 x - 6 x ^ { 2 } = 80$$

Whakaoti mō x (complex solution)

$$x=\frac{7+\sqrt{71}i}{3}\approx 2.333333333+2.808716591i$$

$$x=\frac{-\sqrt{71}i+7}{3}\approx 2.333333333-2.808716591i$$

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.

$$-6x^{2}+28x=80$$

Me tango 80 mai i ngā taha e rua o te whārite.

$$-6x^{2}+28x-80=80-80$$

Mā te tango i te 80 i a ia ake anō ka toe ko te 0.

$$-6x^{2}+28x-80=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -6 mō a, 28 mō b, me -80 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-28±\sqrt{28^{2}-4\left(-6\right)\left(-80\right)}}{2\left(-6\right)}$$

Pūrua 28.

$$x=\frac{-28±\sqrt{784-4\left(-6\right)\left(-80\right)}}{2\left(-6\right)}$$

Whakareatia -4 ki te -6.

$$x=\frac{-28±\sqrt{784+24\left(-80\right)}}{2\left(-6\right)}$$

Whakareatia 24 ki te -80.

$$x=\frac{-28±\sqrt{784-1920}}{2\left(-6\right)}$$

Tāpiri 784 ki te -1920.

$$x=\frac{-28±\sqrt{-1136}}{2\left(-6\right)}$$

Tuhia te pūtakerua o te -1136.

$$x=\frac{-28±4\sqrt{71}i}{2\left(-6\right)}$$

Whakareatia 2 ki te -6.

$$x=\frac{-28±4\sqrt{71}i}{-12}$$

Nā, me whakaoti te whārite x=\frac{-28±4\sqrt{71}i}{-12} ina he tāpiri te ±. Tāpiri -28 ki te 4i\sqrt{71}.

$$x=\frac{-28+4\sqrt{71}i}{-12}$$

Whakawehe -28+4i\sqrt{71} ki te -12.

$$x=\frac{-\sqrt{71}i+7}{3}$$

Nā, me whakaoti te whārite x=\frac{-28±4\sqrt{71}i}{-12} ina he tango te ±. Tango 4i\sqrt{71} mai i -28.

$$x=\frac{-4\sqrt{71}i-28}{-12}$$

Whakawehe -28-4i\sqrt{71} ki te -12.

$$x=\frac{7+\sqrt{71}i}{3}$$

Kua oti te whārite te whakatau.

$$x=\frac{-\sqrt{71}i+7}{3} x=\frac{7+\sqrt{71}i}{3}$$

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.

$$-6x^{2}+28x=80$$

Whakawehea ngā taha e rua ki te -6.

$$\frac{-6x^{2}+28x}{-6}=\frac{80}{-6}$$

Mā te whakawehe ki te -6 ka wetekia te whakareanga ki te -6.

$$x^{2}+\frac{28}{-6}x=\frac{80}{-6}$$

Whakahekea te hautanga \frac{28}{-6} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x^{2}-\frac{14}{3}x=\frac{80}{-6}$$

Whakahekea te hautanga \frac{80}{-6} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x^{2}-\frac{14}{3}x=-\frac{40}{3}$$

Whakawehea te -\frac{14}{3}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{7}{3}. Nā, tāpiria te pūrua o te -\frac{7}{3} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{14}{3}x+\left(-\frac{7}{3}\right)^{2}=-\frac{40}{3}+\left(-\frac{7}{3}\right)^{2}$$

Pūruatia -\frac{7}{3} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{14}{3}x+\frac{49}{9}=-\frac{40}{3}+\frac{49}{9}$$

Tāpiri -\frac{40}{3} ki te \frac{49}{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.

$$x^{2}-\frac{14}{3}x+\frac{49}{9}=-\frac{71}{9}$$

Tauwehea x^{2}-\frac{14}{3}x+\frac{49}{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}.

$$\left(x-\frac{7}{3}\right)^{2}=-\frac{71}{9}$$

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

$$\sqrt{\left(x-\frac{7}{3}\right)^{2}}=\sqrt{-\frac{71}{9}}$$

Whakarūnātia.

$$x-\frac{7}{3}=\frac{\sqrt{71}i}{3} x-\frac{7}{3}=-\frac{\sqrt{71}i}{3}$$

Me tāpiri \frac{7}{3} ki ngā taha e rua o te whārite.

$$x=\frac{7+\sqrt{71}i}{3} x=\frac{-\sqrt{71}i+7}{3}$$