Whakaoti i te 2x^2-8x-223=0

$$2 x ^ { 2 } - 8 x - 223 = 0$$

Whakaoti mō x

$$x=\frac{\sqrt{462}}{2}+2\approx 12.74709263$$

$$x=-\frac{\sqrt{462}}{2}+2\approx -8.74709263$$

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.

$$2x^{2}-8x-223=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 2 mō a, -8 mō b, me -223 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\times 2\left(-223\right)}}{2\times 2}$$

Pūrua -8.

$$x=\frac{-\left(-8\right)±\sqrt{64-4\times 2\left(-223\right)}}{2\times 2}$$

Whakareatia -4 ki te 2.

$$x=\frac{-\left(-8\right)±\sqrt{64-8\left(-223\right)}}{2\times 2}$$

Whakareatia -8 ki te -223.

$$x=\frac{-\left(-8\right)±\sqrt{64+1784}}{2\times 2}$$

Tāpiri 64 ki te 1784.

$$x=\frac{-\left(-8\right)±\sqrt{1848}}{2\times 2}$$

Tuhia te pūtakerua o te 1848.

$$x=\frac{-\left(-8\right)±2\sqrt{462}}{2\times 2}$$

Ko te tauaro o -8 ko 8.

$$x=\frac{8±2\sqrt{462}}{2\times 2}$$

Whakareatia 2 ki te 2.

$$x=\frac{8±2\sqrt{462}}{4}$$

Nā, me whakaoti te whārite x=\frac{8±2\sqrt{462}}{4} ina he tāpiri te ±. Tāpiri 8 ki te 2\sqrt{462}.

$$x=\frac{2\sqrt{462}+8}{4}$$

Whakawehe 8+2\sqrt{462} ki te 4.

$$x=\frac{\sqrt{462}}{2}+2$$

Nā, me whakaoti te whārite x=\frac{8±2\sqrt{462}}{4} ina he tango te ±. Tango 2\sqrt{462} mai i 8.

$$x=\frac{8-2\sqrt{462}}{4}$$

Whakawehe 8-2\sqrt{462} ki te 4.

$$x=-\frac{\sqrt{462}}{2}+2$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{462}}{2}+2 x=-\frac{\sqrt{462}}{2}+2$$

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.

$$2x^{2}-8x-223=0$$

Me tāpiri 223 ki ngā taha e rua o te whārite.

$$2x^{2}-8x-223-\left(-223\right)=-\left(-223\right)$$

Mā te tango i te -223 i a ia ake anō ka toe ko te 0.

$$2x^{2}-8x=-\left(-223\right)$$

Tango -223 mai i 0.

$$2x^{2}-8x=223$$

Whakawehea ngā taha e rua ki te 2.

$$\frac{2x^{2}-8x}{2}=\frac{223}{2}$$

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

$$x^{2}+\left(-\frac{8}{2}\right)x=\frac{223}{2}$$

Whakawehe -8 ki te 2.

$$x^{2}-4x=\frac{223}{2}$$

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

$$x^{2}-4x+\left(-2\right)^{2}=\frac{223}{2}+\left(-2\right)^{2}$$

Pūrua -2.

$$x^{2}-4x+4=\frac{223}{2}+4$$

Tāpiri \frac{223}{2} ki te 4.

$$x^{2}-4x+4=\frac{231}{2}$$

Tauwehea x^{2}-4x+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}.

$$\left(x-2\right)^{2}=\frac{231}{2}$$

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

$$\sqrt{\left(x-2\right)^{2}}=\sqrt{\frac{231}{2}}$$

Whakarūnātia.

$$x-2=\frac{\sqrt{462}}{2} x-2=-\frac{\sqrt{462}}{2}$$

Me tāpiri 2 ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{462}}{2}+2 x=-\frac{\sqrt{462}}{2}+2$$