Whakaoti i te 46x^2-6x*3+3=0

$$46 { x }^{ 2 } -6x \times 3+3=0$$

Whakaoti mō x (complex solution)

$$x=\frac{9+\sqrt{57}i}{46}\approx 0.195652174+0.164126836i$$

$$x=\frac{-\sqrt{57}i+9}{46}\approx 0.195652174-0.164126836i$$

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.

$$46x^{2}-18x+3=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 46 mō a, -18 mō b, me 3 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-18\right)±\sqrt{\left(-18\right)^{2}-4\times 46\times 3}}{2\times 46}$$

Pūrua -18.

$$x=\frac{-\left(-18\right)±\sqrt{324-4\times 46\times 3}}{2\times 46}$$

Whakareatia -4 ki te 46.

$$x=\frac{-\left(-18\right)±\sqrt{324-184\times 3}}{2\times 46}$$

Whakareatia -184 ki te 3.

$$x=\frac{-\left(-18\right)±\sqrt{324-552}}{2\times 46}$$

Tāpiri 324 ki te -552.

$$x=\frac{-\left(-18\right)±\sqrt{-228}}{2\times 46}$$

Tuhia te pūtakerua o te -228.

$$x=\frac{-\left(-18\right)±2\sqrt{57}i}{2\times 46}$$

Ko te tauaro o -18 ko 18.

$$x=\frac{18±2\sqrt{57}i}{2\times 46}$$

Whakareatia 2 ki te 46.

$$x=\frac{18±2\sqrt{57}i}{92}$$

Nā, me whakaoti te whārite x=\frac{18±2\sqrt{57}i}{92} ina he tāpiri te ±. Tāpiri 18 ki te 2i\sqrt{57}.

$$x=\frac{18+2\sqrt{57}i}{92}$$

Whakawehe 18+2i\sqrt{57} ki te 92.

$$x=\frac{9+\sqrt{57}i}{46}$$

Nā, me whakaoti te whārite x=\frac{18±2\sqrt{57}i}{92} ina he tango te ±. Tango 2i\sqrt{57} mai i 18.

$$x=\frac{-2\sqrt{57}i+18}{92}$$

Whakawehe 18-2i\sqrt{57} ki te 92.

$$x=\frac{-\sqrt{57}i+9}{46}$$

Kua oti te whārite te whakatau.

$$x=\frac{9+\sqrt{57}i}{46} x=\frac{-\sqrt{57}i+9}{46}$$

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.

$$46x^{2}-18x+3=0$$

Me tango 3 mai i ngā taha e rua o te whārite.

$$46x^{2}-18x+3-3=-3$$

Mā te tango i te 3 i a ia ake anō ka toe ko te 0.

$$46x^{2}-18x=-3$$

Whakawehea ngā taha e rua ki te 46.

$$\frac{46x^{2}-18x}{46}=-\frac{3}{46}$$

Mā te whakawehe ki te 46 ka wetekia te whakareanga ki te 46.

$$x^{2}+\left(-\frac{18}{46}\right)x=-\frac{3}{46}$$

Whakahekea te hautanga \frac{-18}{46} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x^{2}-\frac{9}{23}x=-\frac{3}{46}$$

Whakawehea te -\frac{9}{23}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{9}{46}. Nā, tāpiria te pūrua o te -\frac{9}{46} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{9}{23}x+\left(-\frac{9}{46}\right)^{2}=-\frac{3}{46}+\left(-\frac{9}{46}\right)^{2}$$

Pūruatia -\frac{9}{46} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{9}{23}x+\frac{81}{2116}=-\frac{3}{46}+\frac{81}{2116}$$

Tāpiri -\frac{3}{46} ki te \frac{81}{2116} 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.

$$x^{2}-\frac{9}{23}x+\frac{81}{2116}=-\frac{57}{2116}$$

Tauwehea x^{2}-\frac{9}{23}x+\frac{81}{2116}. 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}.

$$\left(x-\frac{9}{46}\right)^{2}=-\frac{57}{2116}$$

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

$$\sqrt{\left(x-\frac{9}{46}\right)^{2}}=\sqrt{-\frac{57}{2116}}$$

Whakarūnātia.

$$x-\frac{9}{46}=\frac{\sqrt{57}i}{46} x-\frac{9}{46}=-\frac{\sqrt{57}i}{46}$$

Me tāpiri \frac{9}{46} ki ngā taha e rua o te whārite.

$$x=\frac{9+\sqrt{57}i}{46} x=\frac{-\sqrt{57}i+9}{46}$$