Whakaoti i te 7x^2-4x+6=0

$$7 x ^ { 2 } - 4 x + 6 = 0$$

Whakaoti mō x (complex solution)

$$x=\frac{2+\sqrt{38}i}{7}\approx 0.285714286+0.880630572i$$

$$x=\frac{-\sqrt{38}i+2}{7}\approx 0.285714286-0.880630572i$$

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.

$$7x^{2}-4x+6=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 7 mō a, -4 mō b, me 6 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\times 7\times 6}}{2\times 7}$$

Pūrua -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\times 7\times 6}}{2\times 7}$$

Whakareatia -4 ki te 7.

$$x=\frac{-\left(-4\right)±\sqrt{16-28\times 6}}{2\times 7}$$

Whakareatia -28 ki te 6.

$$x=\frac{-\left(-4\right)±\sqrt{16-168}}{2\times 7}$$

Tāpiri 16 ki te -168.

$$x=\frac{-\left(-4\right)±\sqrt{-152}}{2\times 7}$$

Tuhia te pūtakerua o te -152.

$$x=\frac{-\left(-4\right)±2\sqrt{38}i}{2\times 7}$$

Ko te tauaro o -4 ko 4.

$$x=\frac{4±2\sqrt{38}i}{2\times 7}$$

Whakareatia 2 ki te 7.

$$x=\frac{4±2\sqrt{38}i}{14}$$

Nā, me whakaoti te whārite x=\frac{4±2\sqrt{38}i}{14} ina he tāpiri te ±. Tāpiri 4 ki te 2i\sqrt{38}.

$$x=\frac{4+2\sqrt{38}i}{14}$$

Whakawehe 4+2i\sqrt{38} ki te 14.

$$x=\frac{2+\sqrt{38}i}{7}$$

Nā, me whakaoti te whārite x=\frac{4±2\sqrt{38}i}{14} ina he tango te ±. Tango 2i\sqrt{38} mai i 4.

$$x=\frac{-2\sqrt{38}i+4}{14}$$

Whakawehe 4-2i\sqrt{38} ki te 14.

$$x=\frac{-\sqrt{38}i+2}{7}$$

Kua oti te whārite te whakatau.

$$x=\frac{2+\sqrt{38}i}{7} x=\frac{-\sqrt{38}i+2}{7}$$

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.

$$7x^{2}-4x+6=0$$

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

$$7x^{2}-4x+6-6=-6$$

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

$$7x^{2}-4x=-6$$

Whakawehea ngā taha e rua ki te 7.

$$\frac{7x^{2}-4x}{7}=-\frac{6}{7}$$

Mā te whakawehe ki te 7 ka wetekia te whakareanga ki te 7.

$$x^{2}-\frac{4}{7}x=-\frac{6}{7}$$

Whakawehea te -\frac{4}{7}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{2}{7}. Nā, tāpiria te pūrua o te -\frac{2}{7} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{4}{7}x+\left(-\frac{2}{7}\right)^{2}=-\frac{6}{7}+\left(-\frac{2}{7}\right)^{2}$$

Pūruatia -\frac{2}{7} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{4}{7}x+\frac{4}{49}=-\frac{6}{7}+\frac{4}{49}$$

Tāpiri -\frac{6}{7} ki te \frac{4}{49} 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.

$$x^{2}-\frac{4}{7}x+\frac{4}{49}=-\frac{38}{49}$$

Tauwehea x^{2}-\frac{4}{7}x+\frac{4}{49}. 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}.

$$\left(x-\frac{2}{7}\right)^{2}=-\frac{38}{49}$$

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

$$\sqrt{\left(x-\frac{2}{7}\right)^{2}}=\sqrt{-\frac{38}{49}}$$

Whakarūnātia.

$$x-\frac{2}{7}=\frac{\sqrt{38}i}{7} x-\frac{2}{7}=-\frac{\sqrt{38}i}{7}$$

Me tāpiri \frac{2}{7} ki ngā taha e rua o te whārite.

$$x=\frac{2+\sqrt{38}i}{7} x=\frac{-\sqrt{38}i+2}{7}$$