Whakaoti i te 2x^2-4x-135=0

$$2 { x }^{ 2 } -4x-135=0$$

Whakaoti mō x

$$x=\frac{\sqrt{274}}{2}+1\approx 9.276472679$$

$$x=-\frac{\sqrt{274}}{2}+1\approx -7.276472679$$

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.

$$2x^{2}-4x-135=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 2 mō a, -4 mō b, me -135 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\times 2\left(-135\right)}}{2\times 2}$$

Pūrua -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\times 2\left(-135\right)}}{2\times 2}$$

Whakareatia -4 ki te 2.

$$x=\frac{-\left(-4\right)±\sqrt{16-8\left(-135\right)}}{2\times 2}$$

Whakareatia -8 ki te -135.

$$x=\frac{-\left(-4\right)±\sqrt{16+1080}}{2\times 2}$$

Tāpiri 16 ki te 1080.

$$x=\frac{-\left(-4\right)±\sqrt{1096}}{2\times 2}$$

Tuhia te pūtakerua o te 1096.

$$x=\frac{-\left(-4\right)±2\sqrt{274}}{2\times 2}$$

Ko te tauaro o -4 ko 4.

$$x=\frac{4±2\sqrt{274}}{2\times 2}$$

Whakareatia 2 ki te 2.

$$x=\frac{4±2\sqrt{274}}{4}$$

Nā, me whakaoti te whārite x=\frac{4±2\sqrt{274}}{4} ina he tāpiri te ±. Tāpiri 4 ki te 2\sqrt{274}.

$$x=\frac{2\sqrt{274}+4}{4}$$

Whakawehe 4+2\sqrt{274} ki te 4.

$$x=\frac{\sqrt{274}}{2}+1$$

Nā, me whakaoti te whārite x=\frac{4±2\sqrt{274}}{4} ina he tango te ±. Tango 2\sqrt{274} mai i 4.

$$x=\frac{4-2\sqrt{274}}{4}$$

Whakawehe 4-2\sqrt{274} ki te 4.

$$x=-\frac{\sqrt{274}}{2}+1$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{274}}{2}+1 x=-\frac{\sqrt{274}}{2}+1$$

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.

$$2x^{2}-4x-135=0$$

Me tāpiri 135 ki ngā taha e rua o te whārite.

$$2x^{2}-4x-135-\left(-135\right)=-\left(-135\right)$$

Mā te tango i te -135 i a ia ake anō ka toe ko te 0.

$$2x^{2}-4x=-\left(-135\right)$$

Tango -135 mai i 0.

$$2x^{2}-4x=135$$

Whakawehea ngā taha e rua ki te 2.

$$\frac{2x^{2}-4x}{2}=\frac{135}{2}$$

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

$$x^{2}+\left(-\frac{4}{2}\right)x=\frac{135}{2}$$

Whakawehe -4 ki te 2.

$$x^{2}-2x=\frac{135}{2}$$

Whakawehea te -2, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -1. Nā, tāpiria te pūrua o te -1 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-2x+1=\frac{135}{2}+1$$

Tāpiri \frac{135}{2} ki te 1.

$$x^{2}-2x+1=\frac{137}{2}$$

Tauwehea x^{2}-2x+1. 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}.

$$\left(x-1\right)^{2}=\frac{137}{2}$$

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

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{\frac{137}{2}}$$

Whakarūnātia.

$$x-1=\frac{\sqrt{274}}{2} x-1=-\frac{\sqrt{274}}{2}$$

Me tāpiri 1 ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{274}}{2}+1 x=-\frac{\sqrt{274}}{2}+1$$