Whakaoti i te 2x^2-40x+1140=0

$$2 { x }^{ 2 } -40x+1140 = 0$$

Whakaoti mō x (complex solution)

$$x=10+\sqrt{470}i\approx 10+21.679483389i$$

$$x=-\sqrt{470}i+10\approx 10-21.679483389i$$

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.

$$2x^{2}-40x+1140=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 2 mō a, -40 mō b, me 1140 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-40\right)±\sqrt{\left(-40\right)^{2}-4\times 2\times 1140}}{2\times 2}$$

Pūrua -40.

$$x=\frac{-\left(-40\right)±\sqrt{1600-4\times 2\times 1140}}{2\times 2}$$

Whakareatia -4 ki te 2.

$$x=\frac{-\left(-40\right)±\sqrt{1600-8\times 1140}}{2\times 2}$$

Whakareatia -8 ki te 1140.

$$x=\frac{-\left(-40\right)±\sqrt{1600-9120}}{2\times 2}$$

Tāpiri 1600 ki te -9120.

$$x=\frac{-\left(-40\right)±\sqrt{-7520}}{2\times 2}$$

Tuhia te pūtakerua o te -7520.

$$x=\frac{-\left(-40\right)±4\sqrt{470}i}{2\times 2}$$

Ko te tauaro o -40 ko 40.

$$x=\frac{40±4\sqrt{470}i}{2\times 2}$$

Whakareatia 2 ki te 2.

$$x=\frac{40±4\sqrt{470}i}{4}$$

Nā, me whakaoti te whārite x=\frac{40±4\sqrt{470}i}{4} ina he tāpiri te ±. Tāpiri 40 ki te 4i\sqrt{470}.

$$x=\frac{40+4\sqrt{470}i}{4}$$

Whakawehe 40+4i\sqrt{470} ki te 4.

$$x=10+\sqrt{470}i$$

Nā, me whakaoti te whārite x=\frac{40±4\sqrt{470}i}{4} ina he tango te ±. Tango 4i\sqrt{470} mai i 40.

$$x=\frac{-4\sqrt{470}i+40}{4}$$

Whakawehe 40-4i\sqrt{470} ki te 4.

$$x=-\sqrt{470}i+10$$

Kua oti te whārite te whakatau.

$$x=10+\sqrt{470}i x=-\sqrt{470}i+10$$

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.

$$2x^{2}-40x+1140=0$$

Me tango 1140 mai i ngā taha e rua o te whārite.

$$2x^{2}-40x+1140-1140=-1140$$

Mā te tango i te 1140 i a ia ake anō ka toe ko te 0.

$$2x^{2}-40x=-1140$$

Whakawehea ngā taha e rua ki te 2.

$$\frac{2x^{2}-40x}{2}=-\frac{1140}{2}$$

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

$$x^{2}+\left(-\frac{40}{2}\right)x=-\frac{1140}{2}$$

Whakawehe -40 ki te 2.

$$x^{2}-20x=-\frac{1140}{2}$$

Whakawehe -1140 ki te 2.

$$x^{2}-20x=-570$$

Whakawehea te -20, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -10. Nā, tāpiria te pūrua o te -10 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-20x+\left(-10\right)^{2}=-570+\left(-10\right)^{2}$$

Pūrua -10.

$$x^{2}-20x+100=-570+100$$

Tāpiri -570 ki te 100.

$$x^{2}-20x+100=-470$$

Tauwehea x^{2}-20x+100. 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}.

$$\left(x-10\right)^{2}=-470$$

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

$$\sqrt{\left(x-10\right)^{2}}=\sqrt{-470}$$

Whakarūnātia.

$$x-10=\sqrt{470}i x-10=-\sqrt{470}i$$

Me tāpiri 10 ki ngā taha e rua o te whārite.

$$x=10+\sqrt{470}i x=-\sqrt{470}i+10$$