Whakaoti i te b^2-8b+27=0

$$b ^ { 2 } - 8 b + 27 = 0$$

Whakaoti mō b

$$b=4+\sqrt{11}i\approx 4+3.31662479i$$

$$b=-\sqrt{11}i+4\approx 4-3.31662479i$$

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.

$$b^{2}-8b+27=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -8 mō b, me 27 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$b=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\times 27}}{2}$$

Pūrua -8.

$$b=\frac{-\left(-8\right)±\sqrt{64-4\times 27}}{2}$$

Whakareatia -4 ki te 27.

$$b=\frac{-\left(-8\right)±\sqrt{64-108}}{2}$$

Tāpiri 64 ki te -108.

$$b=\frac{-\left(-8\right)±\sqrt{-44}}{2}$$

Tuhia te pūtakerua o te -44.

$$b=\frac{-\left(-8\right)±2\sqrt{11}i}{2}$$

Ko te tauaro o -8 ko 8.

$$b=\frac{8±2\sqrt{11}i}{2}$$

Nā, me whakaoti te whārite b=\frac{8±2\sqrt{11}i}{2} ina he tāpiri te ±. Tāpiri 8 ki te 2i\sqrt{11}.

$$b=\frac{8+2\sqrt{11}i}{2}$$

Whakawehe 8+2i\sqrt{11} ki te 2.

$$b=4+\sqrt{11}i$$

Nā, me whakaoti te whārite b=\frac{8±2\sqrt{11}i}{2} ina he tango te ±. Tango 2i\sqrt{11} mai i 8.

$$b=\frac{-2\sqrt{11}i+8}{2}$$

Whakawehe 8-2i\sqrt{11} ki te 2.

$$b=-\sqrt{11}i+4$$

Kua oti te whārite te whakatau.

$$b=4+\sqrt{11}i b=-\sqrt{11}i+4$$

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.

$$b^{2}-8b+27=0$$

Me tango 27 mai i ngā taha e rua o te whārite.

$$b^{2}-8b+27-27=-27$$

Mā te tango i te 27 i a ia ake anō ka toe ko te 0.

$$b^{2}-8b=-27$$

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.

$$b^{2}-8b+\left(-4\right)^{2}=-27+\left(-4\right)^{2}$$

Pūrua -4.

$$b^{2}-8b+16=-27+16$$

Tāpiri -27 ki te 16.

$$b^{2}-8b+16=-11$$

Tauwehea b^{2}-8b+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}.

$$\left(b-4\right)^{2}=-11$$

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

$$\sqrt{\left(b-4\right)^{2}}=\sqrt{-11}$$

Whakarūnātia.

$$b-4=\sqrt{11}i b-4=-\sqrt{11}i$$

Me tāpiri 4 ki ngā taha e rua o te whārite.

$$b=4+\sqrt{11}i b=-\sqrt{11}i+4$$