Whakaoti i te z^2-2iz+3=0

$$z ^ { 2 } - 2 i z + 3 = 0$$

Whakaoti mō z

$$z=3i$$

$$z=-i$$

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.

$$z^{2}-2iz+3=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -2i mō b, me 3 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$z=\frac{2i±\sqrt{\left(-2i\right)^{2}-4\times 3}}{2}$$

Pūrua -2i.

$$z=\frac{2i±\sqrt{-4-4\times 3}}{2}$$

Whakareatia -4 ki te 3.

$$z=\frac{2i±\sqrt{-4-12}}{2}$$

Tāpiri -4 ki te -12.

$$z=\frac{2i±\sqrt{-16}}{2}$$

Tuhia te pūtakerua o te -16.

$$z=\frac{2i±4i}{2}$$

Nā, me whakaoti te whārite z=\frac{2i±4i}{2} ina he tāpiri te ±. Tāpiri 2i ki te 4i.

$$z=\frac{6i}{2}$$

Whakawehe 6i ki te 2.

$$z=3i$$

Nā, me whakaoti te whārite z=\frac{2i±4i}{2} ina he tango te ±. Tango 4i mai i 2i.

$$z=\frac{-2i}{2}$$

Whakawehe -2i ki te 2.

$$z=-i$$

Kua oti te whārite te whakatau.

$$z=3i z=-i$$

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.

$$z^{2}-2iz+3=0$$

Me tango 3 mai i ngā taha e rua o te whārite.

$$z^{2}-2iz+3-3=-3$$

Mā te tango i te 3 i a ia ake anō ka toe ko te 0.

$$z^{2}-2iz=-3$$

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

$$z^{2}-2iz+\left(-i\right)^{2}=-3+\left(-i\right)^{2}$$

Pūrua -i.

$$z^{2}-2iz-1=-3-1$$

Tāpiri -3 ki te -1.

$$z^{2}-2iz-1=-4$$

Tauwehea z^{2}-2iz-1. 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}.

$$\left(z-i\right)^{2}=-4$$

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

$$\sqrt{\left(z-i\right)^{2}}=\sqrt{-4}$$

Whakarūnātia.

$$z-i=2i z-i=-2i$$

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

$$z=3i z=-i$$