Whakaoti i te 12x^2-2x+5=0

$$12 x ^ { 2 } - 2 x + 5 = 0$$

Whakaoti mō x (complex solution)

$$x=\frac{1+\sqrt{59}i}{12}\approx 0.083333333+0.640095479i$$

$$x=\frac{-\sqrt{59}i+1}{12}\approx 0.083333333-0.640095479i$$

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.

$$12x^{2}-2x+5=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 12 mō a, -2 mō b, me 5 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\times 12\times 5}}{2\times 12}$$

Pūrua -2.

$$x=\frac{-\left(-2\right)±\sqrt{4-4\times 12\times 5}}{2\times 12}$$

Whakareatia -4 ki te 12.

$$x=\frac{-\left(-2\right)±\sqrt{4-48\times 5}}{2\times 12}$$

Whakareatia -48 ki te 5.

$$x=\frac{-\left(-2\right)±\sqrt{4-240}}{2\times 12}$$

Tāpiri 4 ki te -240.

$$x=\frac{-\left(-2\right)±\sqrt{-236}}{2\times 12}$$

Tuhia te pūtakerua o te -236.

$$x=\frac{-\left(-2\right)±2\sqrt{59}i}{2\times 12}$$

Ko te tauaro o -2 ko 2.

$$x=\frac{2±2\sqrt{59}i}{2\times 12}$$

Whakareatia 2 ki te 12.

$$x=\frac{2±2\sqrt{59}i}{24}$$

Nā, me whakaoti te whārite x=\frac{2±2\sqrt{59}i}{24} ina he tāpiri te ±. Tāpiri 2 ki te 2i\sqrt{59}.

$$x=\frac{2+2\sqrt{59}i}{24}$$

Whakawehe 2+2i\sqrt{59} ki te 24.

$$x=\frac{1+\sqrt{59}i}{12}$$

Nā, me whakaoti te whārite x=\frac{2±2\sqrt{59}i}{24} ina he tango te ±. Tango 2i\sqrt{59} mai i 2.

$$x=\frac{-2\sqrt{59}i+2}{24}$$

Whakawehe 2-2i\sqrt{59} ki te 24.

$$x=\frac{-\sqrt{59}i+1}{12}$$

Kua oti te whārite te whakatau.

$$x=\frac{1+\sqrt{59}i}{12} x=\frac{-\sqrt{59}i+1}{12}$$

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.

$$12x^{2}-2x+5=0$$

Me tango 5 mai i ngā taha e rua o te whārite.

$$12x^{2}-2x+5-5=-5$$

Mā te tango i te 5 i a ia ake anō ka toe ko te 0.

$$12x^{2}-2x=-5$$

Whakawehea ngā taha e rua ki te 12.

$$\frac{12x^{2}-2x}{12}=-\frac{5}{12}$$

Mā te whakawehe ki te 12 ka wetekia te whakareanga ki te 12.

$$x^{2}+\left(-\frac{2}{12}\right)x=-\frac{5}{12}$$

Whakahekea te hautanga \frac{-2}{12} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x^{2}-\frac{1}{6}x=-\frac{5}{12}$$

Whakawehea te -\frac{1}{6}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{1}{12}. Nā, tāpiria te pūrua o te -\frac{1}{12} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{1}{6}x+\left(-\frac{1}{12}\right)^{2}=-\frac{5}{12}+\left(-\frac{1}{12}\right)^{2}$$

Pūruatia -\frac{1}{12} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{1}{6}x+\frac{1}{144}=-\frac{5}{12}+\frac{1}{144}$$

Tāpiri -\frac{5}{12} ki te \frac{1}{144} 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.

$$x^{2}-\frac{1}{6}x+\frac{1}{144}=-\frac{59}{144}$$

Tauwehea x^{2}-\frac{1}{6}x+\frac{1}{144}. 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}.

$$\left(x-\frac{1}{12}\right)^{2}=-\frac{59}{144}$$

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

$$\sqrt{\left(x-\frac{1}{12}\right)^{2}}=\sqrt{-\frac{59}{144}}$$

Whakarūnātia.

$$x-\frac{1}{12}=\frac{\sqrt{59}i}{12} x-\frac{1}{12}=-\frac{\sqrt{59}i}{12}$$

Me tāpiri \frac{1}{12} ki ngā taha e rua o te whārite.

$$x=\frac{1+\sqrt{59}i}{12} x=\frac{-\sqrt{59}i+1}{12}$$