Whakaoti i te 2x^2-28x+171=0

$$2 x ^ { 2 } - 28 x + 171 = 0$$

Whakaoti mō x (complex solution)

$$x=\frac{\sqrt{146}i}{2}+7\approx 7+6.041522987i$$

$$x=-\frac{\sqrt{146}i}{2}+7\approx 7-6.041522987i$$

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.

$$2x^{2}-28x+171=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 2 mō a, -28 mō b, me 171 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-28\right)±\sqrt{\left(-28\right)^{2}-4\times 2\times 171}}{2\times 2}$$

Pūrua -28.

$$x=\frac{-\left(-28\right)±\sqrt{784-4\times 2\times 171}}{2\times 2}$$

Whakareatia -4 ki te 2.

$$x=\frac{-\left(-28\right)±\sqrt{784-8\times 171}}{2\times 2}$$

Whakareatia -8 ki te 171.

$$x=\frac{-\left(-28\right)±\sqrt{784-1368}}{2\times 2}$$

Tāpiri 784 ki te -1368.

$$x=\frac{-\left(-28\right)±\sqrt{-584}}{2\times 2}$$

Tuhia te pūtakerua o te -584.

$$x=\frac{-\left(-28\right)±2\sqrt{146}i}{2\times 2}$$

Ko te tauaro o -28 ko 28.

$$x=\frac{28±2\sqrt{146}i}{2\times 2}$$

Whakareatia 2 ki te 2.

$$x=\frac{28±2\sqrt{146}i}{4}$$

Nā, me whakaoti te whārite x=\frac{28±2\sqrt{146}i}{4} ina he tāpiri te ±. Tāpiri 28 ki te 2i\sqrt{146}.

$$x=\frac{28+2\sqrt{146}i}{4}$$

Whakawehe 28+2i\sqrt{146} ki te 4.

$$x=\frac{\sqrt{146}i}{2}+7$$

Nā, me whakaoti te whārite x=\frac{28±2\sqrt{146}i}{4} ina he tango te ±. Tango 2i\sqrt{146} mai i 28.

$$x=\frac{-2\sqrt{146}i+28}{4}$$

Whakawehe 28-2i\sqrt{146} ki te 4.

$$x=-\frac{\sqrt{146}i}{2}+7$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{146}i}{2}+7 x=-\frac{\sqrt{146}i}{2}+7$$

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.

$$2x^{2}-28x+171=0$$

Me tango 171 mai i ngā taha e rua o te whārite.

$$2x^{2}-28x+171-171=-171$$

Mā te tango i te 171 i a ia ake anō ka toe ko te 0.

$$2x^{2}-28x=-171$$

Whakawehea ngā taha e rua ki te 2.

$$\frac{2x^{2}-28x}{2}=-\frac{171}{2}$$

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

$$x^{2}+\left(-\frac{28}{2}\right)x=-\frac{171}{2}$$

Whakawehe -28 ki te 2.

$$x^{2}-14x=-\frac{171}{2}$$

Whakawehea te -14, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -7. Nā, tāpiria te pūrua o te -7 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-14x+\left(-7\right)^{2}=-\frac{171}{2}+\left(-7\right)^{2}$$

Pūrua -7.

$$x^{2}-14x+49=-\frac{171}{2}+49$$

Tāpiri -\frac{171}{2} ki te 49.

$$x^{2}-14x+49=-\frac{73}{2}$$

Tauwehea x^{2}-14x+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}.

$$\left(x-7\right)^{2}=-\frac{73}{2}$$

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

$$\sqrt{\left(x-7\right)^{2}}=\sqrt{-\frac{73}{2}}$$

Whakarūnātia.

$$x-7=\frac{\sqrt{146}i}{2} x-7=-\frac{\sqrt{146}i}{2}$$

Me tāpiri 7 ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{146}i}{2}+7 x=-\frac{\sqrt{146}i}{2}+7$$