Whakaoti i te 2x^2+16x-1=0

$$2 { x }^{ 2 } +16x-1=0$$

Whakaoti mō x

$$x=\frac{\sqrt{66}}{2}-4\approx 0.062019202$$

$$x=-\frac{\sqrt{66}}{2}-4\approx -8.062019202$$

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.

$$2x^{2}+16x-1=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 2 mō a, 16 mō b, me -1 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-16±\sqrt{16^{2}-4\times 2\left(-1\right)}}{2\times 2}$$

Pūrua 16.

$$x=\frac{-16±\sqrt{256-4\times 2\left(-1\right)}}{2\times 2}$$

Whakareatia -4 ki te 2.

$$x=\frac{-16±\sqrt{256-8\left(-1\right)}}{2\times 2}$$

Whakareatia -8 ki te -1.

$$x=\frac{-16±\sqrt{256+8}}{2\times 2}$$

Tāpiri 256 ki te 8.

$$x=\frac{-16±\sqrt{264}}{2\times 2}$$

Tuhia te pūtakerua o te 264.

$$x=\frac{-16±2\sqrt{66}}{2\times 2}$$

Whakareatia 2 ki te 2.

$$x=\frac{-16±2\sqrt{66}}{4}$$

Nā, me whakaoti te whārite x=\frac{-16±2\sqrt{66}}{4} ina he tāpiri te ±. Tāpiri -16 ki te 2\sqrt{66}.

$$x=\frac{2\sqrt{66}-16}{4}$$

Whakawehe -16+2\sqrt{66} ki te 4.

$$x=\frac{\sqrt{66}}{2}-4$$

Nā, me whakaoti te whārite x=\frac{-16±2\sqrt{66}}{4} ina he tango te ±. Tango 2\sqrt{66} mai i -16.

$$x=\frac{-2\sqrt{66}-16}{4}$$

Whakawehe -16-2\sqrt{66} ki te 4.

$$x=-\frac{\sqrt{66}}{2}-4$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{66}}{2}-4 x=-\frac{\sqrt{66}}{2}-4$$

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.

$$2x^{2}+16x-1=0$$

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

$$2x^{2}+16x-1-\left(-1\right)=-\left(-1\right)$$

Mā te tango i te -1 i a ia ake anō ka toe ko te 0.

$$2x^{2}+16x=-\left(-1\right)$$

Tango -1 mai i 0.

$$2x^{2}+16x=1$$

Whakawehea ngā taha e rua ki te 2.

$$\frac{2x^{2}+16x}{2}=\frac{1}{2}$$

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

$$x^{2}+\frac{16}{2}x=\frac{1}{2}$$

Whakawehe 16 ki te 2.

$$x^{2}+8x=\frac{1}{2}$$

Whakawehea te 8, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 4. Nā, tāpiria te pūrua o te 4 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+8x+4^{2}=\frac{1}{2}+4^{2}$$

Pūrua 4.

$$x^{2}+8x+16=\frac{1}{2}+16$$

Tāpiri \frac{1}{2} ki te 16.

$$x^{2}+8x+16=\frac{33}{2}$$

Tauwehea x^{2}+8x+16. 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}.

$$\left(x+4\right)^{2}=\frac{33}{2}$$

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

$$\sqrt{\left(x+4\right)^{2}}=\sqrt{\frac{33}{2}}$$

Whakarūnātia.

$$x+4=\frac{\sqrt{66}}{2} x+4=-\frac{\sqrt{66}}{2}$$

Me tango 4 mai i ngā taha e rua o te whārite.

$$x=\frac{\sqrt{66}}{2}-4 x=-\frac{\sqrt{66}}{2}-4$$