Whakaoti i te -2x^2+7x+6=0

$$- 2 x ^ { 2 } + 7 x + 6 = 0$$

Whakaoti mō x

$$x = \frac{\sqrt{97} + 7}{4} \approx 4.21221445$$

$$x=\frac{7-\sqrt{97}}{4}\approx -0.71221445$$

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.

$$-2x^{2}+7x+6=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -2 mō a, 7 mō b, me 6 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-7±\sqrt{7^{2}-4\left(-2\right)\times 6}}{2\left(-2\right)}$$

Pūrua 7.

$$x=\frac{-7±\sqrt{49-4\left(-2\right)\times 6}}{2\left(-2\right)}$$

Whakareatia -4 ki te -2.

$$x=\frac{-7±\sqrt{49+8\times 6}}{2\left(-2\right)}$$

Whakareatia 8 ki te 6.

$$x=\frac{-7±\sqrt{49+48}}{2\left(-2\right)}$$

Tāpiri 49 ki te 48.

$$x=\frac{-7±\sqrt{97}}{2\left(-2\right)}$$

Whakareatia 2 ki te -2.

$$x=\frac{-7±\sqrt{97}}{-4}$$

Nā, me whakaoti te whārite x=\frac{-7±\sqrt{97}}{-4} ina he tāpiri te ±. Tāpiri -7 ki te \sqrt{97}.

$$x=\frac{\sqrt{97}-7}{-4}$$

Whakawehe -7+\sqrt{97} ki te -4.

$$x=\frac{7-\sqrt{97}}{4}$$

Nā, me whakaoti te whārite x=\frac{-7±\sqrt{97}}{-4} ina he tango te ±. Tango \sqrt{97} mai i -7.

$$x=\frac{-\sqrt{97}-7}{-4}$$

Whakawehe -7-\sqrt{97} ki te -4.

$$x=\frac{\sqrt{97}+7}{4}$$

Kua oti te whārite te whakatau.

$$x=\frac{7-\sqrt{97}}{4} x=\frac{\sqrt{97}+7}{4}$$

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.

$$-2x^{2}+7x+6=0$$

Me tango 6 mai i ngā taha e rua o te whārite.

$$-2x^{2}+7x+6-6=-6$$

Mā te tango i te 6 i a ia ake anō ka toe ko te 0.

$$-2x^{2}+7x=-6$$

Whakawehea ngā taha e rua ki te -2.

$$\frac{-2x^{2}+7x}{-2}=-\frac{6}{-2}$$

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

$$x^{2}+\frac{7}{-2}x=-\frac{6}{-2}$$

Whakawehe 7 ki te -2.

$$x^{2}-\frac{7}{2}x=-\frac{6}{-2}$$

Whakawehe -6 ki te -2.

$$x^{2}-\frac{7}{2}x=3$$

Whakawehea te -\frac{7}{2}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{7}{4}. Nā, tāpiria te pūrua o te -\frac{7}{4} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{7}{2}x+\left(-\frac{7}{4}\right)^{2}=3+\left(-\frac{7}{4}\right)^{2}$$

Pūruatia -\frac{7}{4} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{7}{2}x+\frac{49}{16}=3+\frac{49}{16}$$

Tāpiri 3 ki te \frac{49}{16}.

$$x^{2}-\frac{7}{2}x+\frac{49}{16}=\frac{97}{16}$$

Tauwehea x^{2}-\frac{7}{2}x+\frac{49}{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}.

$$\left(x-\frac{7}{4}\right)^{2}=\frac{97}{16}$$

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

$$\sqrt{\left(x-\frac{7}{4}\right)^{2}}=\sqrt{\frac{97}{16}}$$

Whakarūnātia.

$$x-\frac{7}{4}=\frac{\sqrt{97}}{4} x-\frac{7}{4}=-\frac{\sqrt{97}}{4}$$

Me tāpiri \frac{7}{4} ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{97}+7}{4} x=\frac{7-\sqrt{97}}{4}$$