Whakaoti i te 2x^2+8x+14=0

$$2 x ^ { 2 } + 8 x + 14 = 0$$

Whakaoti mō x (complex solution)

$$x=-2+\sqrt{3}i\approx -2+1.732050808i$$

$$x=-\sqrt{3}i-2\approx -2-1.732050808i$$

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.

$$2x^{2}+8x+14=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 2 mō a, 8 mō b, me 14 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-8±\sqrt{8^{2}-4\times 2\times 14}}{2\times 2}$$

Pūrua 8.

$$x=\frac{-8±\sqrt{64-4\times 2\times 14}}{2\times 2}$$

Whakareatia -4 ki te 2.

$$x=\frac{-8±\sqrt{64-8\times 14}}{2\times 2}$$

Whakareatia -8 ki te 14.

$$x=\frac{-8±\sqrt{64-112}}{2\times 2}$$

Tāpiri 64 ki te -112.

$$x=\frac{-8±\sqrt{-48}}{2\times 2}$$

Tuhia te pūtakerua o te -48.

$$x=\frac{-8±4\sqrt{3}i}{2\times 2}$$

Whakareatia 2 ki te 2.

$$x=\frac{-8±4\sqrt{3}i}{4}$$

Nā, me whakaoti te whārite x=\frac{-8±4\sqrt{3}i}{4} ina he tāpiri te ±. Tāpiri -8 ki te 4i\sqrt{3}.

$$x=\frac{-8+4\sqrt{3}i}{4}$$

Whakawehe -8+4i\sqrt{3} ki te 4.

$$x=-2+\sqrt{3}i$$

Nā, me whakaoti te whārite x=\frac{-8±4\sqrt{3}i}{4} ina he tango te ±. Tango 4i\sqrt{3} mai i -8.

$$x=\frac{-4\sqrt{3}i-8}{4}$$

Whakawehe -8-4i\sqrt{3} ki te 4.

$$x=-\sqrt{3}i-2$$

Kua oti te whārite te whakatau.

$$x=-2+\sqrt{3}i x=-\sqrt{3}i-2$$

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.

$$2x^{2}+8x+14=0$$

Me tango 14 mai i ngā taha e rua o te whārite.

$$2x^{2}+8x+14-14=-14$$

Mā te tango i te 14 i a ia ake anō ka toe ko te 0.

$$2x^{2}+8x=-14$$

Whakawehea ngā taha e rua ki te 2.

$$\frac{2x^{2}+8x}{2}=-\frac{14}{2}$$

Mā te whakawehe ki te 2 ka wetekia te whakareanga ki te 2.

$$x^{2}+\frac{8}{2}x=-\frac{14}{2}$$

Whakawehe 8 ki te 2.

$$x^{2}+4x=-\frac{14}{2}$$

Whakawehe -14 ki te 2.

$$x^{2}+4x=-7$$

Whakawehea te 4, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 2. Nā, tāpiria te pūrua o te 2 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+4x+2^{2}=-7+2^{2}$$

Pūrua 2.

$$x^{2}+4x+4=-7+4$$

Tāpiri -7 ki te 4.

$$x^{2}+4x+4=-3$$

Tauwehea x^{2}+4x+4. 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}.

$$\left(x+2\right)^{2}=-3$$

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

$$\sqrt{\left(x+2\right)^{2}}=\sqrt{-3}$$

Whakarūnātia.

$$x+2=\sqrt{3}i x+2=-\sqrt{3}i$$

Me tango 2 mai i ngā taha e rua o te whārite.

$$x=-2+\sqrt{3}i x=-\sqrt{3}i-2$$