Whakaoti i te 2x^2-11x+16=0

$$2 x ^ { 2 } - 11 x + 16 = 0$$

Whakaoti mō x (complex solution)

$$x=\frac{11+\sqrt{7}i}{4}\approx 2.75+0.661437828i$$

$$x=\frac{-\sqrt{7}i+11}{4}\approx 2.75-0.661437828i$$

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.

$$2x^{2}-11x+16=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 2 mō a, -11 mō b, me 16 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-11\right)±\sqrt{\left(-11\right)^{2}-4\times 2\times 16}}{2\times 2}$$

Pūrua -11.

$$x=\frac{-\left(-11\right)±\sqrt{121-4\times 2\times 16}}{2\times 2}$$

Whakareatia -4 ki te 2.

$$x=\frac{-\left(-11\right)±\sqrt{121-8\times 16}}{2\times 2}$$

Whakareatia -8 ki te 16.

$$x=\frac{-\left(-11\right)±\sqrt{121-128}}{2\times 2}$$

Tāpiri 121 ki te -128.

$$x=\frac{-\left(-11\right)±\sqrt{-7}}{2\times 2}$$

Tuhia te pūtakerua o te -7.

$$x=\frac{-\left(-11\right)±\sqrt{7}i}{2\times 2}$$

Ko te tauaro o -11 ko 11.

$$x=\frac{11±\sqrt{7}i}{2\times 2}$$

Whakareatia 2 ki te 2.

$$x=\frac{11±\sqrt{7}i}{4}$$

Nā, me whakaoti te whārite x=\frac{11±\sqrt{7}i}{4} ina he tāpiri te ±. Tāpiri 11 ki te i\sqrt{7}.

$$x=\frac{11+\sqrt{7}i}{4}$$

Nā, me whakaoti te whārite x=\frac{11±\sqrt{7}i}{4} ina he tango te ±. Tango i\sqrt{7} mai i 11.

$$x=\frac{-\sqrt{7}i+11}{4}$$

Kua oti te whārite te whakatau.

$$x=\frac{11+\sqrt{7}i}{4} x=\frac{-\sqrt{7}i+11}{4}$$

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.

$$2x^{2}-11x+16=0$$

Me tango 16 mai i ngā taha e rua o te whārite.

$$2x^{2}-11x+16-16=-16$$

Mā te tango i te 16 i a ia ake anō ka toe ko te 0.

$$2x^{2}-11x=-16$$

Whakawehea ngā taha e rua ki te 2.

$$\frac{2x^{2}-11x}{2}=-\frac{16}{2}$$

Mā te whakawehe ki te 2 ka wetekia te whakareanga ki te 2.

$$x^{2}-\frac{11}{2}x=-\frac{16}{2}$$

Whakawehe -16 ki te 2.

$$x^{2}-\frac{11}{2}x=-8$$

Whakawehea te -\frac{11}{2}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{11}{4}. Nā, tāpiria te pūrua o te -\frac{11}{4} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{11}{2}x+\left(-\frac{11}{4}\right)^{2}=-8+\left(-\frac{11}{4}\right)^{2}$$

Pūruatia -\frac{11}{4} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{11}{2}x+\frac{121}{16}=-8+\frac{121}{16}$$

Tāpiri -8 ki te \frac{121}{16}.

$$x^{2}-\frac{11}{2}x+\frac{121}{16}=-\frac{7}{16}$$

Tauwehea x^{2}-\frac{11}{2}x+\frac{121}{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}.

$$\left(x-\frac{11}{4}\right)^{2}=-\frac{7}{16}$$

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

$$\sqrt{\left(x-\frac{11}{4}\right)^{2}}=\sqrt{-\frac{7}{16}}$$

Whakarūnātia.

$$x-\frac{11}{4}=\frac{\sqrt{7}i}{4} x-\frac{11}{4}=-\frac{\sqrt{7}i}{4}$$

Me tāpiri \frac{11}{4} ki ngā taha e rua o te whārite.

$$x=\frac{11+\sqrt{7}i}{4} x=\frac{-\sqrt{7}i+11}{4}$$