Whakaoti i te 3x^2+6x=-18

$$3 x ^ { 2 } + 6 x = - 18$$

Whakaoti mō x (complex solution)

$$x=-1+\sqrt{5}i\approx -1+2.236067977i$$

$$x=-\sqrt{5}i-1\approx -1-2.236067977i$$

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.

$$3x^{2}+6x=-18$$

Me tāpiri 18 ki ngā taha e rua o te whārite.

$$3x^{2}+6x-\left(-18\right)=-18-\left(-18\right)$$

Mā te tango i te -18 i a ia ake anō ka toe ko te 0.

$$3x^{2}+6x-\left(-18\right)=0$$

Tango -18 mai i 0.

$$3x^{2}+6x+18=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 3 mō a, 6 mō b, me 18 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-6±\sqrt{6^{2}-4\times 3\times 18}}{2\times 3}$$

Pūrua 6.

$$x=\frac{-6±\sqrt{36-4\times 3\times 18}}{2\times 3}$$

Whakareatia -4 ki te 3.

$$x=\frac{-6±\sqrt{36-12\times 18}}{2\times 3}$$

Whakareatia -12 ki te 18.

$$x=\frac{-6±\sqrt{36-216}}{2\times 3}$$

Tāpiri 36 ki te -216.

$$x=\frac{-6±\sqrt{-180}}{2\times 3}$$

Tuhia te pūtakerua o te -180.

$$x=\frac{-6±6\sqrt{5}i}{2\times 3}$$

Whakareatia 2 ki te 3.

$$x=\frac{-6±6\sqrt{5}i}{6}$$

Nā, me whakaoti te whārite x=\frac{-6±6\sqrt{5}i}{6} ina he tāpiri te ±. Tāpiri -6 ki te 6i\sqrt{5}.

$$x=\frac{-6+6\sqrt{5}i}{6}$$

Whakawehe -6+6i\sqrt{5} ki te 6.

$$x=-1+\sqrt{5}i$$

Nā, me whakaoti te whārite x=\frac{-6±6\sqrt{5}i}{6} ina he tango te ±. Tango 6i\sqrt{5} mai i -6.

$$x=\frac{-6\sqrt{5}i-6}{6}$$

Whakawehe -6-6i\sqrt{5} ki te 6.

$$x=-\sqrt{5}i-1$$

Kua oti te whārite te whakatau.

$$x=-1+\sqrt{5}i x=-\sqrt{5}i-1$$

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.

$$3x^{2}+6x=-18$$

Whakawehea ngā taha e rua ki te 3.

$$\frac{3x^{2}+6x}{3}=-\frac{18}{3}$$

Mā te whakawehe ki te 3 ka wetekia te whakareanga ki te 3.

$$x^{2}+\frac{6}{3}x=-\frac{18}{3}$$

Whakawehe 6 ki te 3.

$$x^{2}+2x=-\frac{18}{3}$$

Whakawehe -18 ki te 3.

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

Whakawehea te 2, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 1. Nā, tāpiria te pūrua o te 1 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+2x+1^{2}=-6+1^{2}$$

Pūrua 1.

$$x^{2}+2x+1=-6+1$$

Tāpiri -6 ki te 1.

$$x^{2}+2x+1=-5$$

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

$$\left(x+1\right)^{2}=-5$$

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

$$\sqrt{\left(x+1\right)^{2}}=\sqrt{-5}$$

Whakarūnātia.

$$x+1=\sqrt{5}i x+1=-\sqrt{5}i$$

Me tango 1 mai i ngā taha e rua o te whārite.

$$x=-1+\sqrt{5}i x=-\sqrt{5}i-1$$