Whakaoti i te x^2+18x+3840=0

$${ x }^{ 2 } +18x+3840=0$$

Whakaoti mō x (complex solution)

$$x=-9+\sqrt{3759}i\approx -9+61.310684224i$$

$$x=-\sqrt{3759}i-9\approx -9-61.310684224i$$

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.

$$x^{2}+18x+3840=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, 18 mō b, me 3840 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-18±\sqrt{18^{2}-4\times 3840}}{2}$$

Pūrua 18.

$$x=\frac{-18±\sqrt{324-4\times 3840}}{2}$$

Whakareatia -4 ki te 3840.

$$x=\frac{-18±\sqrt{324-15360}}{2}$$

Tāpiri 324 ki te -15360.

$$x=\frac{-18±\sqrt{-15036}}{2}$$

Tuhia te pūtakerua o te -15036.

$$x=\frac{-18±2\sqrt{3759}i}{2}$$

Nā, me whakaoti te whārite x=\frac{-18±2\sqrt{3759}i}{2} ina he tāpiri te ±. Tāpiri -18 ki te 2i\sqrt{3759}.

$$x=\frac{-18+2\sqrt{3759}i}{2}$$

Whakawehe -18+2i\sqrt{3759} ki te 2.

$$x=-9+\sqrt{3759}i$$

Nā, me whakaoti te whārite x=\frac{-18±2\sqrt{3759}i}{2} ina he tango te ±. Tango 2i\sqrt{3759} mai i -18.

$$x=\frac{-2\sqrt{3759}i-18}{2}$$

Whakawehe -18-2i\sqrt{3759} ki te 2.

$$x=-\sqrt{3759}i-9$$

Kua oti te whārite te whakatau.

$$x=-9+\sqrt{3759}i x=-\sqrt{3759}i-9$$

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.

$$x^{2}+18x+3840=0$$

Me tango 3840 mai i ngā taha e rua o te whārite.

$$x^{2}+18x+3840-3840=-3840$$

Mā te tango i te 3840 i a ia ake anō ka toe ko te 0.

$$x^{2}+18x=-3840$$

Whakawehea te 18, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 9. Nā, tāpiria te pūrua o te 9 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+18x+9^{2}=-3840+9^{2}$$

Pūrua 9.

$$x^{2}+18x+81=-3840+81$$

Tāpiri -3840 ki te 81.

$$x^{2}+18x+81=-3759$$

Tauwehea x^{2}+18x+81. 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}.

$$\left(x+9\right)^{2}=-3759$$

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

$$\sqrt{\left(x+9\right)^{2}}=\sqrt{-3759}$$

Whakarūnātia.

$$x+9=\sqrt{3759}i x+9=-\sqrt{3759}i$$

Me tango 9 mai i ngā taha e rua o te whārite.

$$x=-9+\sqrt{3759}i x=-\sqrt{3759}i-9$$