Whakaoti i te left(3xright)^2-4x+1=0

$${ \left(3x \right) }^{ 2 } -4x+1=0$$

Whakaoti mō x (complex solution)

$$x=\frac{2+\sqrt{5}i}{9}\approx 0.222222222+0.248451997i$$

$$x=\frac{-\sqrt{5}i+2}{9}\approx 0.222222222-0.248451997i$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakarohaina te \left(3x\right)^{2}.

$$3^{2}x^{2}-4x+1=0$$

Tātaihia te 3 mā te pū o 2, kia riro ko 9.

$$9x^{2}-4x+1=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 9 mō a, -4 mō b, me 1 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\times 9}}{2\times 9}$$

Pūrua -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\times 9}}{2\times 9}$$

Whakareatia -4 ki te 9.

$$x=\frac{-\left(-4\right)±\sqrt{16-36}}{2\times 9}$$

Tāpiri 16 ki te -36.

$$x=\frac{-\left(-4\right)±\sqrt{-20}}{2\times 9}$$

Tuhia te pūtakerua o te -20.

$$x=\frac{-\left(-4\right)±2\sqrt{5}i}{2\times 9}$$

Ko te tauaro o -4 ko 4.

$$x=\frac{4±2\sqrt{5}i}{2\times 9}$$

Whakareatia 2 ki te 9.

$$x=\frac{4±2\sqrt{5}i}{18}$$

Nā, me whakaoti te whārite x=\frac{4±2\sqrt{5}i}{18} ina he tāpiri te ±. Tāpiri 4 ki te 2i\sqrt{5}.

$$x=\frac{4+2\sqrt{5}i}{18}$$

Whakawehe 4+2i\sqrt{5} ki te 18.

$$x=\frac{2+\sqrt{5}i}{9}$$

Nā, me whakaoti te whārite x=\frac{4±2\sqrt{5}i}{18} ina he tango te ±. Tango 2i\sqrt{5} mai i 4.

$$x=\frac{-2\sqrt{5}i+4}{18}$$

Whakawehe 4-2i\sqrt{5} ki te 18.

$$x=\frac{-\sqrt{5}i+2}{9}$$

Kua oti te whārite te whakatau.

$$x=\frac{2+\sqrt{5}i}{9} x=\frac{-\sqrt{5}i+2}{9}$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakarohaina te \left(3x\right)^{2}.

$$3^{2}x^{2}-4x+1=0$$

Tātaihia te 3 mā te pū o 2, kia riro ko 9.

$$9x^{2}-4x+1=0$$

Tangohia te 1 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$9x^{2}-4x=-1$$

Whakawehea ngā taha e rua ki te 9.

$$\frac{9x^{2}-4x}{9}=-\frac{1}{9}$$

Mā te whakawehe ki te 9 ka wetekia te whakareanga ki te 9.

$$x^{2}-\frac{4}{9}x=-\frac{1}{9}$$

Whakawehea te -\frac{4}{9}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{2}{9}. Nā, tāpiria te pūrua o te -\frac{2}{9} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{4}{9}x+\left(-\frac{2}{9}\right)^{2}=-\frac{1}{9}+\left(-\frac{2}{9}\right)^{2}$$

Pūruatia -\frac{2}{9} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{4}{9}x+\frac{4}{81}=-\frac{1}{9}+\frac{4}{81}$$

Tāpiri -\frac{1}{9} ki te \frac{4}{81} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x^{2}-\frac{4}{9}x+\frac{4}{81}=-\frac{5}{81}$$

Tauwehea x^{2}-\frac{4}{9}x+\frac{4}{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}.

$$\left(x-\frac{2}{9}\right)^{2}=-\frac{5}{81}$$

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

$$\sqrt{\left(x-\frac{2}{9}\right)^{2}}=\sqrt{-\frac{5}{81}}$$

Whakarūnātia.

$$x-\frac{2}{9}=\frac{\sqrt{5}i}{9} x-\frac{2}{9}=-\frac{\sqrt{5}i}{9}$$

Me tāpiri \frac{2}{9} ki ngā taha e rua o te whārite.

$$x=\frac{2+\sqrt{5}i}{9} x=\frac{-\sqrt{5}i+2}{9}$$