Whakaoti i te 4x^2-2x+7=0

$$4 { x }^{ 2 } -2x+7=0$$

Whakaoti mō x (complex solution)

$$x=\frac{1+3\sqrt{3}i}{4}\approx 0.25+1.299038106i$$

$$x=\frac{-3\sqrt{3}i+1}{4}\approx 0.25-1.299038106i$$

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.

$$4x^{2}-2x+7=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 4 mō a, -2 mō b, me 7 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\times 4\times 7}}{2\times 4}$$

Pūrua -2.

$$x=\frac{-\left(-2\right)±\sqrt{4-4\times 4\times 7}}{2\times 4}$$

Whakareatia -4 ki te 4.

$$x=\frac{-\left(-2\right)±\sqrt{4-16\times 7}}{2\times 4}$$

Whakareatia -16 ki te 7.

$$x=\frac{-\left(-2\right)±\sqrt{4-112}}{2\times 4}$$

Tāpiri 4 ki te -112.

$$x=\frac{-\left(-2\right)±\sqrt{-108}}{2\times 4}$$

Tuhia te pūtakerua o te -108.

$$x=\frac{-\left(-2\right)±6\sqrt{3}i}{2\times 4}$$

Ko te tauaro o -2 ko 2.

$$x=\frac{2±6\sqrt{3}i}{2\times 4}$$

Whakareatia 2 ki te 4.

$$x=\frac{2±6\sqrt{3}i}{8}$$

Nā, me whakaoti te whārite x=\frac{2±6\sqrt{3}i}{8} ina he tāpiri te ±. Tāpiri 2 ki te 6i\sqrt{3}.

$$x=\frac{2+6\sqrt{3}i}{8}$$

Whakawehe 2+6i\sqrt{3} ki te 8.

$$x=\frac{1+3\sqrt{3}i}{4}$$

Nā, me whakaoti te whārite x=\frac{2±6\sqrt{3}i}{8} ina he tango te ±. Tango 6i\sqrt{3} mai i 2.

$$x=\frac{-6\sqrt{3}i+2}{8}$$

Whakawehe 2-6i\sqrt{3} ki te 8.

$$x=\frac{-3\sqrt{3}i+1}{4}$$

Kua oti te whārite te whakatau.

$$x=\frac{1+3\sqrt{3}i}{4} x=\frac{-3\sqrt{3}i+1}{4}$$

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.

$$4x^{2}-2x+7=0$$

Me tango 7 mai i ngā taha e rua o te whārite.

$$4x^{2}-2x+7-7=-7$$

Mā te tango i te 7 i a ia ake anō ka toe ko te 0.

$$4x^{2}-2x=-7$$

Whakawehea ngā taha e rua ki te 4.

$$\frac{4x^{2}-2x}{4}=-\frac{7}{4}$$

Mā te whakawehe ki te 4 ka wetekia te whakareanga ki te 4.

$$x^{2}+\left(-\frac{2}{4}\right)x=-\frac{7}{4}$$

Whakahekea te hautanga \frac{-2}{4} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x^{2}-\frac{1}{2}x=-\frac{7}{4}$$

Whakawehea te -\frac{1}{2}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{1}{4}. Nā, tāpiria te pūrua o te -\frac{1}{4} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{1}{2}x+\left(-\frac{1}{4}\right)^{2}=-\frac{7}{4}+\left(-\frac{1}{4}\right)^{2}$$

Pūruatia -\frac{1}{4} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{1}{2}x+\frac{1}{16}=-\frac{7}{4}+\frac{1}{16}$$

Tāpiri -\frac{7}{4} ki te \frac{1}{16} 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.

$$x^{2}-\frac{1}{2}x+\frac{1}{16}=-\frac{27}{16}$$

Tauwehea x^{2}-\frac{1}{2}x+\frac{1}{16}. 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}.

$$\left(x-\frac{1}{4}\right)^{2}=-\frac{27}{16}$$

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

$$\sqrt{\left(x-\frac{1}{4}\right)^{2}}=\sqrt{-\frac{27}{16}}$$

Whakarūnātia.

$$x-\frac{1}{4}=\frac{3\sqrt{3}i}{4} x-\frac{1}{4}=-\frac{3\sqrt{3}i}{4}$$

Me tāpiri \frac{1}{4} ki ngā taha e rua o te whārite.

$$x=\frac{1+3\sqrt{3}i}{4} x=\frac{-3\sqrt{3}i+1}{4}$$