Whakaoti i te x^2-6x+11=0

$${ x }^{ 2 } -6x+11 = 0$$

Whakaoti mō x (complex solution)

$$x=3+\sqrt{2}i\approx 3+1.414213562i$$

$$x=-\sqrt{2}i+3\approx 3-1.414213562i$$

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.

$$x^{2}-6x+11=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -6 mō b, me 11 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\times 11}}{2}$$

Pūrua -6.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\times 11}}{2}$$

Whakareatia -4 ki te 11.

$$x=\frac{-\left(-6\right)±\sqrt{36-44}}{2}$$

Tāpiri 36 ki te -44.

$$x=\frac{-\left(-6\right)±\sqrt{-8}}{2}$$

Tuhia te pūtakerua o te -8.

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

Ko te tauaro o -6 ko 6.

$$x=\frac{6±2\sqrt{2}i}{2}$$

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

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

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

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

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

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

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

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

Kua oti te whārite te whakatau.

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

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.

$$x^{2}-6x+11=0$$

Me tango 11 mai i ngā taha e rua o te whārite.

$$x^{2}-6x+11-11=-11$$

Mā te tango i te 11 i a ia ake anō ka toe ko te 0.

$$x^{2}-6x=-11$$

Whakawehea te -6, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -3. Nā, tāpiria te pūrua o te -3 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-6x+\left(-3\right)^{2}=-11+\left(-3\right)^{2}$$

Pūrua -3.

$$x^{2}-6x+9=-11+9$$

Tāpiri -11 ki te 9.

$$x^{2}-6x+9=-2$$

Tauwehea te x^{2}-6x+9. Ko te tikanga, ina ko x^{2}+bx+c he pūrua tika, ka taea te tauwehe i ngā wā katoa hei \left(x+\frac{b}{2}\right)^{2}.

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

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

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

Whakarūnātia.

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

Me tāpiri 3 ki ngā taha e rua o te whārite.

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