Whakaoti i te -2x^2+136x+1800=350

$$- 2 x ^ { 2 } + 136 x + 1800 = 350$$

Whakaoti mō x

$$x=3\sqrt{209}+34\approx 77.370496884$$

$$x=34-3\sqrt{209}\approx -9.370496884$$

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.

$$-2x^{2}+136x+1800=350$$

Me tango 350 mai i ngā taha e rua o te whārite.

$$-2x^{2}+136x+1800-350=350-350$$

Mā te tango i te 350 i a ia ake anō ka toe ko te 0.

$$-2x^{2}+136x+1800-350=0$$

Tango 350 mai i 1800.

$$-2x^{2}+136x+1450=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -2 mō a, 136 mō b, me 1450 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-136±\sqrt{136^{2}-4\left(-2\right)\times 1450}}{2\left(-2\right)}$$

Pūrua 136.

$$x=\frac{-136±\sqrt{18496-4\left(-2\right)\times 1450}}{2\left(-2\right)}$$

Whakareatia -4 ki te -2.

$$x=\frac{-136±\sqrt{18496+8\times 1450}}{2\left(-2\right)}$$

Whakareatia 8 ki te 1450.

$$x=\frac{-136±\sqrt{18496+11600}}{2\left(-2\right)}$$

Tāpiri 18496 ki te 11600.

$$x=\frac{-136±\sqrt{30096}}{2\left(-2\right)}$$

Tuhia te pūtakerua o te 30096.

$$x=\frac{-136±12\sqrt{209}}{2\left(-2\right)}$$

Whakareatia 2 ki te -2.

$$x=\frac{-136±12\sqrt{209}}{-4}$$

Nā, me whakaoti te whārite x=\frac{-136±12\sqrt{209}}{-4} ina he tāpiri te ±. Tāpiri -136 ki te 12\sqrt{209}.

$$x=\frac{12\sqrt{209}-136}{-4}$$

Whakawehe -136+12\sqrt{209} ki te -4.

$$x=34-3\sqrt{209}$$

Nā, me whakaoti te whārite x=\frac{-136±12\sqrt{209}}{-4} ina he tango te ±. Tango 12\sqrt{209} mai i -136.

$$x=\frac{-12\sqrt{209}-136}{-4}$$

Whakawehe -136-12\sqrt{209} ki te -4.

$$x=3\sqrt{209}+34$$

Kua oti te whārite te whakatau.

$$x=34-3\sqrt{209} x=3\sqrt{209}+34$$

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.

$$-2x^{2}+136x+1800=350$$

Me tango 1800 mai i ngā taha e rua o te whārite.

$$-2x^{2}+136x+1800-1800=350-1800$$

Mā te tango i te 1800 i a ia ake anō ka toe ko te 0.

$$-2x^{2}+136x=350-1800$$

Tango 1800 mai i 350.

$$-2x^{2}+136x=-1450$$

Whakawehea ngā taha e rua ki te -2.

$$\frac{-2x^{2}+136x}{-2}=-\frac{1450}{-2}$$

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

$$x^{2}+\frac{136}{-2}x=-\frac{1450}{-2}$$

Whakawehe 136 ki te -2.

$$x^{2}-68x=-\frac{1450}{-2}$$

Whakawehe -1450 ki te -2.

$$x^{2}-68x=725$$

Whakawehea te -68, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -34. Nā, tāpiria te pūrua o te -34 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-68x+\left(-34\right)^{2}=725+\left(-34\right)^{2}$$

Pūrua -34.

$$x^{2}-68x+1156=725+1156$$

Tāpiri 725 ki te 1156.

$$x^{2}-68x+1156=1881$$

Tauwehea x^{2}-68x+1156. 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}.

$$\left(x-34\right)^{2}=1881$$

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

$$\sqrt{\left(x-34\right)^{2}}=\sqrt{1881}$$

Whakarūnātia.

$$x-34=3\sqrt{209} x-34=-3\sqrt{209}$$

Me tāpiri 34 ki ngā taha e rua o te whārite.

$$x=3\sqrt{209}+34 x=34-3\sqrt{209}$$