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

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

Whakaoti mō x

$$x = \frac{\sqrt{65} + 1}{4} \approx 2.265564437$$

$$x=\frac{1-\sqrt{65}}{4}\approx -1.765564437$$

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.

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

Me tāpiri 8 ki ngā taha e rua o te whārite.

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

Mā te tango i te -8 i a ia ake anō ka toe ko te 0.

$$-2x^{2}+x-\left(-8\right)=0$$

Tango -8 mai i 0.

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

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -2 mō a, 1 mō b, me 8 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1±\sqrt{1^{2}-4\left(-2\right)\times 8}}{2\left(-2\right)}$$

Pūrua 1.

$$x=\frac{-1±\sqrt{1-4\left(-2\right)\times 8}}{2\left(-2\right)}$$

Whakareatia -4 ki te -2.

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

Whakareatia 8 ki te 8.

$$x=\frac{-1±\sqrt{1+64}}{2\left(-2\right)}$$

Tāpiri 1 ki te 64.

$$x=\frac{-1±\sqrt{65}}{2\left(-2\right)}$$

Whakareatia 2 ki te -2.

$$x=\frac{-1±\sqrt{65}}{-4}$$

Nā, me whakaoti te whārite x=\frac{-1±\sqrt{65}}{-4} ina he tāpiri te ±. Tāpiri -1 ki te \sqrt{65}.

$$x=\frac{\sqrt{65}-1}{-4}$$

Whakawehe -1+\sqrt{65} ki te -4.

$$x=\frac{1-\sqrt{65}}{4}$$

Nā, me whakaoti te whārite x=\frac{-1±\sqrt{65}}{-4} ina he tango te ±. Tango \sqrt{65} mai i -1.

$$x=\frac{-\sqrt{65}-1}{-4}$$

Whakawehe -1-\sqrt{65} ki te -4.

$$x=\frac{\sqrt{65}+1}{4}$$

Kua oti te whārite te whakatau.

$$x=\frac{1-\sqrt{65}}{4} x=\frac{\sqrt{65}+1}{4}$$

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.

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

Whakawehea ngā taha e rua ki te -2.

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

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

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

Whakawehe 1 ki te -2.

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

Whakawehe -8 ki te -2.

$$x^{2}-\frac{1}{2}x=4$$

Whakawehea te -\frac{1}{2}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{1}{4}. Nā, tāpiria te pūrua o te -\frac{1}{4} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{1}{2}x+\left(-\frac{1}{4}\right)^{2}=4+\left(-\frac{1}{4}\right)^{2}$$

Pūruatia -\frac{1}{4} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{1}{2}x+\frac{1}{16}=4+\frac{1}{16}$$

Tāpiri 4 ki te \frac{1}{16}.

$$x^{2}-\frac{1}{2}x+\frac{1}{16}=\frac{65}{16}$$

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

$$\left(x-\frac{1}{4}\right)^{2}=\frac{65}{16}$$

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

$$\sqrt{\left(x-\frac{1}{4}\right)^{2}}=\sqrt{\frac{65}{16}}$$

Whakarūnātia.

$$x-\frac{1}{4}=\frac{\sqrt{65}}{4} x-\frac{1}{4}=-\frac{\sqrt{65}}{4}$$

Me tāpiri \frac{1}{4} ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{65}+1}{4} x=\frac{1-\sqrt{65}}{4}$$