Whakaoti i te 3x^2-18x+225=6

$$3 x ^ { 2 } - 18 x + 225 = 6$$

Whakaoti mō x (complex solution)

$$x=3+8i$$

$$x=3-8i$$

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.

$$3x^{2}-18x+225=6$$

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

$$3x^{2}-18x+225-6=6-6$$

Mā te tango i te 6 i a ia ake anō ka toe ko te 0.

$$3x^{2}-18x+225-6=0$$

Tango 6 mai i 225.

$$3x^{2}-18x+219=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 3 mō a, -18 mō b, me 219 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-18\right)±\sqrt{\left(-18\right)^{2}-4\times 3\times 219}}{2\times 3}$$

Pūrua -18.

$$x=\frac{-\left(-18\right)±\sqrt{324-4\times 3\times 219}}{2\times 3}$$

Whakareatia -4 ki te 3.

$$x=\frac{-\left(-18\right)±\sqrt{324-12\times 219}}{2\times 3}$$

Whakareatia -12 ki te 219.

$$x=\frac{-\left(-18\right)±\sqrt{324-2628}}{2\times 3}$$

Tāpiri 324 ki te -2628.

$$x=\frac{-\left(-18\right)±\sqrt{-2304}}{2\times 3}$$

Tuhia te pūtakerua o te -2304.

$$x=\frac{-\left(-18\right)±48i}{2\times 3}$$

Ko te tauaro o -18 ko 18.

$$x=\frac{18±48i}{2\times 3}$$

Whakareatia 2 ki te 3.

$$x=\frac{18±48i}{6}$$

Nā, me whakaoti te whārite x=\frac{18±48i}{6} ina he tāpiri te ±. Tāpiri 18 ki te 48i.

$$x=\frac{18+48i}{6}$$

Whakawehe 18+48i ki te 6.

$$x=3+8i$$

Nā, me whakaoti te whārite x=\frac{18±48i}{6} ina he tango te ±. Tango 48i mai i 18.

$$x=\frac{18-48i}{6}$$

Whakawehe 18-48i ki te 6.

$$x=3-8i$$

Kua oti te whārite te whakatau.

$$x=3+8i x=3-8i$$

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.

$$3x^{2}-18x+225=6$$

Me tango 225 mai i ngā taha e rua o te whārite.

$$3x^{2}-18x+225-225=6-225$$

Mā te tango i te 225 i a ia ake anō ka toe ko te 0.

$$3x^{2}-18x=6-225$$

Tango 225 mai i 6.

$$3x^{2}-18x=-219$$

Whakawehea ngā taha e rua ki te 3.

$$\frac{3x^{2}-18x}{3}=-\frac{219}{3}$$

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

$$x^{2}+\left(-\frac{18}{3}\right)x=-\frac{219}{3}$$

Whakawehe -18 ki te 3.

$$x^{2}-6x=-\frac{219}{3}$$

Whakawehe -219 ki te 3.

$$x^{2}-6x=-73$$

Whakawehea te -6, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -3. Nā, tāpiria te pūrua o te -3 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-6x+\left(-3\right)^{2}=-73+\left(-3\right)^{2}$$

Pūrua -3.

$$x^{2}-6x+9=-73+9$$

Tāpiri -73 ki te 9.

$$x^{2}-6x+9=-64$$

Tauwehea x^{2}-6x+9. 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}.

$$\left(x-3\right)^{2}=-64$$

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

$$\sqrt{\left(x-3\right)^{2}}=\sqrt{-64}$$

Whakarūnātia.

$$x-3=8i x-3=-8i$$

Me tāpiri 3 ki ngā taha e rua o te whārite.

$$x=3+8i x=3-8i$$