Whakaoti i te x^2+12x+64=0

$${ x }^{ 2 } +12x+64=0$$

Whakaoti mō x (complex solution)

$$x=-6+2\sqrt{7}i\approx -6+5.291502622i$$

$$x=-2\sqrt{7}i-6\approx -6-5.291502622i$$

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.

$$x^{2}+12x+64=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, 12 mō b, me 64 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-12±\sqrt{12^{2}-4\times 64}}{2}$$

Pūrua 12.

$$x=\frac{-12±\sqrt{144-4\times 64}}{2}$$

Whakareatia -4 ki te 64.

$$x=\frac{-12±\sqrt{144-256}}{2}$$

Tāpiri 144 ki te -256.

$$x=\frac{-12±\sqrt{-112}}{2}$$

Tuhia te pūtakerua o te -112.

$$x=\frac{-12±4\sqrt{7}i}{2}$$

Nā, me whakaoti te whārite x=\frac{-12±4\sqrt{7}i}{2} ina he tāpiri te ±. Tāpiri -12 ki te 4i\sqrt{7}.

$$x=\frac{-12+4\sqrt{7}i}{2}$$

Whakawehe -12+4i\sqrt{7} ki te 2.

$$x=-6+2\sqrt{7}i$$

Nā, me whakaoti te whārite x=\frac{-12±4\sqrt{7}i}{2} ina he tango te ±. Tango 4i\sqrt{7} mai i -12.

$$x=\frac{-4\sqrt{7}i-12}{2}$$

Whakawehe -12-4i\sqrt{7} ki te 2.

$$x=-2\sqrt{7}i-6$$

Kua oti te whārite te whakatau.

$$x=-6+2\sqrt{7}i x=-2\sqrt{7}i-6$$

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.

$$x^{2}+12x+64=0$$

Me tango 64 mai i ngā taha e rua o te whārite.

$$x^{2}+12x+64-64=-64$$

Mā te tango i te 64 i a ia ake anō ka toe ko te 0.

$$x^{2}+12x=-64$$

Whakawehea te 12, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 6. Nā, tāpiria te pūrua o te 6 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+12x+6^{2}=-64+6^{2}$$

Pūrua 6.

$$x^{2}+12x+36=-64+36$$

Tāpiri -64 ki te 36.

$$x^{2}+12x+36=-28$$

Tauwehea x^{2}+12x+36. 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}.

$$\left(x+6\right)^{2}=-28$$

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

$$\sqrt{\left(x+6\right)^{2}}=\sqrt{-28}$$

Whakarūnātia.

$$x+6=2\sqrt{7}i x+6=-2\sqrt{7}i$$

Me tango 6 mai i ngā taha e rua o te whārite.

$$x=-6+2\sqrt{7}i x=-2\sqrt{7}i-6$$