Whakaoti i te -x^2+10x-81=0

$$- { x }^{ 2 } +10x-81=0$$

Whakaoti mō x (complex solution)

$$x=-2\sqrt{14}i+5\approx 5-7.483314774i$$

$$x=5+2\sqrt{14}i\approx 5+7.483314774i$$

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.

$$-x^{2}+10x-81=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -1 mō a, 10 mō b, me -81 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-10±\sqrt{10^{2}-4\left(-1\right)\left(-81\right)}}{2\left(-1\right)}$$

Pūrua 10.

$$x=\frac{-10±\sqrt{100-4\left(-1\right)\left(-81\right)}}{2\left(-1\right)}$$

Whakareatia -4 ki te -1.

$$x=\frac{-10±\sqrt{100+4\left(-81\right)}}{2\left(-1\right)}$$

Whakareatia 4 ki te -81.

$$x=\frac{-10±\sqrt{100-324}}{2\left(-1\right)}$$

Tāpiri 100 ki te -324.

$$x=\frac{-10±\sqrt{-224}}{2\left(-1\right)}$$

Tuhia te pūtakerua o te -224.

$$x=\frac{-10±4\sqrt{14}i}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$x=\frac{-10±4\sqrt{14}i}{-2}$$

Nā, me whakaoti te whārite x=\frac{-10±4\sqrt{14}i}{-2} ina he tāpiri te ±. Tāpiri -10 ki te 4i\sqrt{14}.

$$x=\frac{-10+4\sqrt{14}i}{-2}$$

Whakawehe -10+4i\sqrt{14} ki te -2.

$$x=-2\sqrt{14}i+5$$

Nā, me whakaoti te whārite x=\frac{-10±4\sqrt{14}i}{-2} ina he tango te ±. Tango 4i\sqrt{14} mai i -10.

$$x=\frac{-4\sqrt{14}i-10}{-2}$$

Whakawehe -10-4i\sqrt{14} ki te -2.

$$x=5+2\sqrt{14}i$$

Kua oti te whārite te whakatau.

$$x=-2\sqrt{14}i+5 x=5+2\sqrt{14}i$$

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.

$$-x^{2}+10x-81=0$$

Me tāpiri 81 ki ngā taha e rua o te whārite.

$$-x^{2}+10x-81-\left(-81\right)=-\left(-81\right)$$

Mā te tango i te -81 i a ia ake anō ka toe ko te 0.

$$-x^{2}+10x=-\left(-81\right)$$

Tango -81 mai i 0.

$$-x^{2}+10x=81$$

Whakawehea ngā taha e rua ki te -1.

$$\frac{-x^{2}+10x}{-1}=\frac{81}{-1}$$

Mā te whakawehe ki te -1 ka wetekia te whakareanga ki te -1.

$$x^{2}+\frac{10}{-1}x=\frac{81}{-1}$$

Whakawehe 10 ki te -1.

$$x^{2}-10x=\frac{81}{-1}$$

Whakawehe 81 ki te -1.

$$x^{2}-10x=-81$$

Whakawehea te -10, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -5. Nā, tāpiria te pūrua o te -5 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-10x+\left(-5\right)^{2}=-81+\left(-5\right)^{2}$$

Pūrua -5.

$$x^{2}-10x+25=-81+25$$

Tāpiri -81 ki te 25.

$$x^{2}-10x+25=-56$$

Tauwehea x^{2}-10x+25. 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}.

$$\left(x-5\right)^{2}=-56$$

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

$$\sqrt{\left(x-5\right)^{2}}=\sqrt{-56}$$

Whakarūnātia.

$$x-5=2\sqrt{14}i x-5=-2\sqrt{14}i$$

Me tāpiri 5 ki ngā taha e rua o te whārite.

$$x=5+2\sqrt{14}i x=-2\sqrt{14}i+5$$