Whakaoti i te -2x^2+2x+15=0

$$- 2 x ^ { 2 } + 2 x + 15 = 0$$

Whakaoti mō x

$$x = \frac{\sqrt{31} + 1}{2} \approx 3.283882181$$

$$x=\frac{1-\sqrt{31}}{2}\approx -2.283882181$$

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.

$$-2x^{2}+2x+15=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -2 mō a, 2 mō b, me 15 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2±\sqrt{2^{2}-4\left(-2\right)\times 15}}{2\left(-2\right)}$$

Pūrua 2.

$$x=\frac{-2±\sqrt{4-4\left(-2\right)\times 15}}{2\left(-2\right)}$$

Whakareatia -4 ki te -2.

$$x=\frac{-2±\sqrt{4+8\times 15}}{2\left(-2\right)}$$

Whakareatia 8 ki te 15.

$$x=\frac{-2±\sqrt{4+120}}{2\left(-2\right)}$$

Tāpiri 4 ki te 120.

$$x=\frac{-2±\sqrt{124}}{2\left(-2\right)}$$

Tuhia te pūtakerua o te 124.

$$x=\frac{-2±2\sqrt{31}}{2\left(-2\right)}$$

Whakareatia 2 ki te -2.

$$x=\frac{-2±2\sqrt{31}}{-4}$$

Nā, me whakaoti te whārite x=\frac{-2±2\sqrt{31}}{-4} ina he tāpiri te ±. Tāpiri -2 ki te 2\sqrt{31}.

$$x=\frac{2\sqrt{31}-2}{-4}$$

Whakawehe -2+2\sqrt{31} ki te -4.

$$x=\frac{1-\sqrt{31}}{2}$$

Nā, me whakaoti te whārite x=\frac{-2±2\sqrt{31}}{-4} ina he tango te ±. Tango 2\sqrt{31} mai i -2.

$$x=\frac{-2\sqrt{31}-2}{-4}$$

Whakawehe -2-2\sqrt{31} ki te -4.

$$x=\frac{\sqrt{31}+1}{2}$$

Kua oti te whārite te whakatau.

$$x=\frac{1-\sqrt{31}}{2} x=\frac{\sqrt{31}+1}{2}$$

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.

$$-2x^{2}+2x+15=0$$

Me tango 15 mai i ngā taha e rua o te whārite.

$$-2x^{2}+2x+15-15=-15$$

Mā te tango i te 15 i a ia ake anō ka toe ko te 0.

$$-2x^{2}+2x=-15$$

Whakawehea ngā taha e rua ki te -2.

$$\frac{-2x^{2}+2x}{-2}=-\frac{15}{-2}$$

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

$$x^{2}+\frac{2}{-2}x=-\frac{15}{-2}$$

Whakawehe 2 ki te -2.

$$x^{2}-x=-\frac{15}{-2}$$

Whakawehe -15 ki te -2.

$$x^{2}-x=\frac{15}{2}$$

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

$$x^{2}-x+\left(-\frac{1}{2}\right)^{2}=\frac{15}{2}+\left(-\frac{1}{2}\right)^{2}$$

Pūruatia -\frac{1}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-x+\frac{1}{4}=\frac{15}{2}+\frac{1}{4}$$

Tāpiri \frac{15}{2} ki te \frac{1}{4} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x^{2}-x+\frac{1}{4}=\frac{31}{4}$$

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

$$\left(x-\frac{1}{2}\right)^{2}=\frac{31}{4}$$

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

$$\sqrt{\left(x-\frac{1}{2}\right)^{2}}=\sqrt{\frac{31}{4}}$$

Whakarūnātia.

$$x-\frac{1}{2}=\frac{\sqrt{31}}{2} x-\frac{1}{2}=-\frac{\sqrt{31}}{2}$$

Me tāpiri \frac{1}{2} ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{31}+1}{2} x=\frac{1-\sqrt{31}}{2}$$