Whakaoti i te x^2-3x+4=0

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

Whakaoti mō x (complex solution)

$$x=\frac{3+\sqrt{7}i}{2}\approx 1.5+1.322875656i$$

$$x=\frac{-\sqrt{7}i+3}{2}\approx 1.5-1.322875656i$$

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.

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

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -3 mō b, me 4 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

Pūrua -3.

$$x=\frac{-\left(-3\right)±\sqrt{9-4\times 4}}{2}$$

Whakareatia -4 ki te 4.

$$x=\frac{-\left(-3\right)±\sqrt{9-16}}{2}$$

Tāpiri 9 ki te -16.

$$x=\frac{-\left(-3\right)±\sqrt{-7}}{2}$$

Tuhia te pūtakerua o te -7.

$$x=\frac{-\left(-3\right)±\sqrt{7}i}{2}$$

Ko te tauaro o -3 ko 3.

$$x=\frac{3±\sqrt{7}i}{2}$$

Nā, me whakaoti te whārite x=\frac{3±\sqrt{7}i}{2} ina he tāpiri te ±. Tāpiri 3 ki te i\sqrt{7}.

$$x=\frac{3+\sqrt{7}i}{2}$$

Nā, me whakaoti te whārite x=\frac{3±\sqrt{7}i}{2} ina he tango te ±. Tango i\sqrt{7} mai i 3.

$$x=\frac{-\sqrt{7}i+3}{2}$$

Kua oti te whārite te whakatau.

$$x=\frac{3+\sqrt{7}i}{2} x=\frac{-\sqrt{7}i+3}{2}$$

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.

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

Me tango 4 mai i ngā taha e rua o te whārite.

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

Mā te tango i te 4 i a ia ake anō ka toe ko te 0.

$$x^{2}-3x=-4$$

Whakawehea te -3, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{3}{2}. Nā, tāpiria te pūrua o te -\frac{3}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

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

Pūruatia -\frac{3}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-3x+\frac{9}{4}=-4+\frac{9}{4}$$

Tāpiri -4 ki te \frac{9}{4}.

$$x^{2}-3x+\frac{9}{4}=-\frac{7}{4}$$

Tauwehea x^{2}-3x+\frac{9}{4}. 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}.

$$\left(x-\frac{3}{2}\right)^{2}=-\frac{7}{4}$$

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

$$\sqrt{\left(x-\frac{3}{2}\right)^{2}}=\sqrt{-\frac{7}{4}}$$

Whakarūnātia.

$$x-\frac{3}{2}=\frac{\sqrt{7}i}{2} x-\frac{3}{2}=-\frac{\sqrt{7}i}{2}$$

Me tāpiri \frac{3}{2} ki ngā taha e rua o te whārite.

$$x=\frac{3+\sqrt{7}i}{2} x=\frac{-\sqrt{7}i+3}{2}$$