Whakaoti i te -2a^2-2a-3=-4a^2

$$- 2 a ^ { 2 } - 2 a - 3 = - 4 a ^ { 2 }$$

Whakaoti mō a

$$a = \frac{\sqrt{7} + 1}{2} \approx 1.822875656$$

$$a=\frac{1-\sqrt{7}}{2}\approx -0.822875656$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Me tāpiri te 4a^{2} ki ngā taha e rua.

$$-2a^{2}-2a-3+4a^{2}=0$$

Pahekotia te -2a^{2} me 4a^{2}, ka 2a^{2}.

$$2a^{2}-2a-3=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 2 mō a, -2 mō b, me -3 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\times 2\left(-3\right)}}{2\times 2}$$

Pūrua -2.

$$a=\frac{-\left(-2\right)±\sqrt{4-4\times 2\left(-3\right)}}{2\times 2}$$

Whakareatia -4 ki te 2.

$$a=\frac{-\left(-2\right)±\sqrt{4-8\left(-3\right)}}{2\times 2}$$

Whakareatia -8 ki te -3.

$$a=\frac{-\left(-2\right)±\sqrt{4+24}}{2\times 2}$$

Tāpiri 4 ki te 24.

$$a=\frac{-\left(-2\right)±\sqrt{28}}{2\times 2}$$

Tuhia te pūtakerua o te 28.

$$a=\frac{-\left(-2\right)±2\sqrt{7}}{2\times 2}$$

Ko te tauaro o -2 ko 2.

$$a=\frac{2±2\sqrt{7}}{2\times 2}$$

Whakareatia 2 ki te 2.

$$a=\frac{2±2\sqrt{7}}{4}$$

Nā, me whakaoti te whārite a=\frac{2±2\sqrt{7}}{4} ina he tāpiri te ±. Tāpiri 2 ki te 2\sqrt{7}.

$$a=\frac{2\sqrt{7}+2}{4}$$

Whakawehe 2+2\sqrt{7} ki te 4.

$$a=\frac{\sqrt{7}+1}{2}$$

Nā, me whakaoti te whārite a=\frac{2±2\sqrt{7}}{4} ina he tango te ±. Tango 2\sqrt{7} mai i 2.

$$a=\frac{2-2\sqrt{7}}{4}$$

Whakawehe 2-2\sqrt{7} ki te 4.

$$a=\frac{1-\sqrt{7}}{2}$$

Kua oti te whārite te whakatau.

$$a=\frac{\sqrt{7}+1}{2} a=\frac{1-\sqrt{7}}{2}$$

Ngā Upane mō te Whakaoti i te Pūrua

Me tāpiri te 4a^{2} ki ngā taha e rua.

$$-2a^{2}-2a-3+4a^{2}=0$$

Pahekotia te -2a^{2} me 4a^{2}, ka 2a^{2}.

$$2a^{2}-2a-3=0$$

Me tāpiri te 3 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$2a^{2}-2a=3$$

Whakawehea ngā taha e rua ki te 2.

$$\frac{2a^{2}-2a}{2}=\frac{3}{2}$$

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

$$a^{2}+\left(-\frac{2}{2}\right)a=\frac{3}{2}$$

Whakawehe -2 ki te 2.

$$a^{2}-a=\frac{3}{2}$$

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.

$$a^{2}-a+\left(-\frac{1}{2}\right)^{2}=\frac{3}{2}+\left(-\frac{1}{2}\right)^{2}$$

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

$$a^{2}-a+\frac{1}{4}=\frac{3}{2}+\frac{1}{4}$$

Tāpiri \frac{3}{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.

$$a^{2}-a+\frac{1}{4}=\frac{7}{4}$$

Tauwehea a^{2}-a+\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}.

$$\left(a-\frac{1}{2}\right)^{2}=\frac{7}{4}$$

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

$$\sqrt{\left(a-\frac{1}{2}\right)^{2}}=\sqrt{\frac{7}{4}}$$

Whakarūnātia.

$$a-\frac{1}{2}=\frac{\sqrt{7}}{2} a-\frac{1}{2}=-\frac{\sqrt{7}}{2}$$

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

$$a=\frac{\sqrt{7}+1}{2} a=\frac{1-\sqrt{7}}{2}$$