Whakaoti i te 5x^2-40x+85=0

$$5 x ^ { 2 } - 40 x + 85 = 0$$

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.

$$5x^{2}-40x+85=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 5 mō a, -40 mō b, me 85 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-40\right)±\sqrt{\left(-40\right)^{2}-4\times 5\times 85}}{2\times 5}$$

Pūrua -40.

$$x=\frac{-\left(-40\right)±\sqrt{1600-4\times 5\times 85}}{2\times 5}$$

Whakareatia -4 ki te 5.

$$x=\frac{-\left(-40\right)±\sqrt{1600-20\times 85}}{2\times 5}$$

Whakareatia -20 ki te 85.

$$x=\frac{-\left(-40\right)±\sqrt{1600-1700}}{2\times 5}$$

Tāpiri 1600 ki te -1700.

$$x=\frac{-\left(-40\right)±\sqrt{-100}}{2\times 5}$$

Tuhia te pūtakerua o te -100.

$$x=\frac{-\left(-40\right)±10i}{2\times 5}$$

Ko te tauaro o -40 ko 40.

$$x=\frac{40±10i}{2\times 5}$$

Whakareatia 2 ki te 5.

$$x=\frac{40±10i}{10}$$

Nā, me whakaoti te whārite x=\frac{40±10i}{10} ina he tāpiri te ±. Tāpiri 40 ki te 10i.

$$x=\frac{40+10i}{10}$$

Whakawehe 40+10i ki te 10.

$$x=4+i$$

Nā, me whakaoti te whārite x=\frac{40±10i}{10} ina he tango te ±. Tango 10i mai i 40.

$$x=\frac{40-10i}{10}$$

Whakawehe 40-10i ki te 10.

$$x=4-i$$

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.

$$5x^{2}-40x+85=0$$

Me tango 85 mai i ngā taha e rua o te whārite.

$$5x^{2}-40x+85-85=-85$$

Mā te tango i te 85 i a ia ake anō ka toe ko te 0.

$$5x^{2}-40x=-85$$

Whakawehea ngā taha e rua ki te 5.

$$\frac{5x^{2}-40x}{5}=-\frac{85}{5}$$

Mā te whakawehe ki te 5 ka wetekia te whakareanga ki te 5.

$$x^{2}+\left(-\frac{40}{5}\right)x=-\frac{85}{5}$$

Whakawehe -40 ki te 5.

$$x^{2}-8x=-\frac{85}{5}$$

Whakawehe -85 ki te 5.

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

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.

$$x^{2}-8x+\left(-4\right)^{2}=-17+\left(-4\right)^{2}$$

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 tāpiri 4 ki ngā taha e rua o te whārite.

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