Whakaoti i te 2x^2-34x+20=0

$$2 { x }^{ 2 } -34x+20=0$$

Whakaoti mō x

$$x = \frac{\sqrt{249} + 17}{2} \approx 16.389866919$$

$$x=\frac{17-\sqrt{249}}{2}\approx 0.610133081$$

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.

$$2x^{2}-34x+20=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 2 mō a, -34 mō b, me 20 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-34\right)±\sqrt{\left(-34\right)^{2}-4\times 2\times 20}}{2\times 2}$$

Pūrua -34.

$$x=\frac{-\left(-34\right)±\sqrt{1156-4\times 2\times 20}}{2\times 2}$$

Whakareatia -4 ki te 2.

$$x=\frac{-\left(-34\right)±\sqrt{1156-8\times 20}}{2\times 2}$$

Whakareatia -8 ki te 20.

$$x=\frac{-\left(-34\right)±\sqrt{1156-160}}{2\times 2}$$

Tāpiri 1156 ki te -160.

$$x=\frac{-\left(-34\right)±\sqrt{996}}{2\times 2}$$

Tuhia te pūtakerua o te 996.

$$x=\frac{-\left(-34\right)±2\sqrt{249}}{2\times 2}$$

Ko te tauaro o -34 ko 34.

$$x=\frac{34±2\sqrt{249}}{2\times 2}$$

Whakareatia 2 ki te 2.

$$x=\frac{34±2\sqrt{249}}{4}$$

Nā, me whakaoti te whārite x=\frac{34±2\sqrt{249}}{4} ina he tāpiri te ±. Tāpiri 34 ki te 2\sqrt{249}.

$$x=\frac{2\sqrt{249}+34}{4}$$

Whakawehe 34+2\sqrt{249} ki te 4.

$$x=\frac{\sqrt{249}+17}{2}$$

Nā, me whakaoti te whārite x=\frac{34±2\sqrt{249}}{4} ina he tango te ±. Tango 2\sqrt{249} mai i 34.

$$x=\frac{34-2\sqrt{249}}{4}$$

Whakawehe 34-2\sqrt{249} ki te 4.

$$x=\frac{17-\sqrt{249}}{2}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{249}+17}{2} x=\frac{17-\sqrt{249}}{2}$$

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.

$$2x^{2}-34x+20=0$$

Me tango 20 mai i ngā taha e rua o te whārite.

$$2x^{2}-34x+20-20=-20$$

Mā te tango i te 20 i a ia ake anō ka toe ko te 0.

$$2x^{2}-34x=-20$$

Whakawehea ngā taha e rua ki te 2.

$$\frac{2x^{2}-34x}{2}=-\frac{20}{2}$$

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

$$x^{2}+\left(-\frac{34}{2}\right)x=-\frac{20}{2}$$

Whakawehe -34 ki te 2.

$$x^{2}-17x=-\frac{20}{2}$$

Whakawehe -20 ki te 2.

$$x^{2}-17x=-10$$

Whakawehea te -17, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{17}{2}. Nā, tāpiria te pūrua o te -\frac{17}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-17x+\left(-\frac{17}{2}\right)^{2}=-10+\left(-\frac{17}{2}\right)^{2}$$

Pūruatia -\frac{17}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-17x+\frac{289}{4}=-10+\frac{289}{4}$$

Tāpiri -10 ki te \frac{289}{4}.

$$x^{2}-17x+\frac{289}{4}=\frac{249}{4}$$

Tauwehea x^{2}-17x+\frac{289}{4}. 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}.

$$\left(x-\frac{17}{2}\right)^{2}=\frac{249}{4}$$

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

$$\sqrt{\left(x-\frac{17}{2}\right)^{2}}=\sqrt{\frac{249}{4}}$$

Whakarūnātia.

$$x-\frac{17}{2}=\frac{\sqrt{249}}{2} x-\frac{17}{2}=-\frac{\sqrt{249}}{2}$$

Me tāpiri \frac{17}{2} ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{249}+17}{2} x=\frac{17-\sqrt{249}}{2}$$