Whakaoti i te -2x^2-x-1=0

$$- 2 x ^ { 2 } - x - 1 = 0$$

Whakaoti mō x (complex solution)

$$x=\frac{-\sqrt{7}i-1}{4}\approx -0.25-0.661437828i$$

$$x=\frac{-1+\sqrt{7}i}{4}\approx -0.25+0.661437828i$$

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.

$$-2x^{2}-x-1=0$$

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

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

Whakareatia -4 ki te -2.

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

Whakareatia 8 ki te -1.

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

Tāpiri 1 ki te -8.

$$x=\frac{-\left(-1\right)±\sqrt{-7}}{2\left(-2\right)}$$

Tuhia te pūtakerua o te -7.

$$x=\frac{-\left(-1\right)±\sqrt{7}i}{2\left(-2\right)}$$

Ko te tauaro o -1 ko 1.

$$x=\frac{1±\sqrt{7}i}{2\left(-2\right)}$$

Whakareatia 2 ki te -2.

$$x=\frac{1±\sqrt{7}i}{-4}$$

Nā, me whakaoti te whārite x=\frac{1±\sqrt{7}i}{-4} ina he tāpiri te ±. Tāpiri 1 ki te i\sqrt{7}.

$$x=\frac{1+\sqrt{7}i}{-4}$$

Whakawehe 1+i\sqrt{7} ki te -4.

$$x=\frac{-\sqrt{7}i-1}{4}$$

Nā, me whakaoti te whārite x=\frac{1±\sqrt{7}i}{-4} ina he tango te ±. Tango i\sqrt{7} mai i 1.

$$x=\frac{-\sqrt{7}i+1}{-4}$$

Whakawehe 1-i\sqrt{7} ki te -4.

$$x=\frac{-1+\sqrt{7}i}{4}$$

Kua oti te whārite te whakatau.

$$x=\frac{-\sqrt{7}i-1}{4} x=\frac{-1+\sqrt{7}i}{4}$$

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.

$$-2x^{2}-x-1=0$$

Me tāpiri 1 ki ngā taha e rua o te whārite.

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

Mā te tango i te -1 i a ia ake anō ka toe ko te 0.

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

Tango -1 mai i 0.

$$-2x^{2}-x=1$$

Whakawehea ngā taha e rua ki te -2.

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

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

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

Whakawehe -1 ki te -2.

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

Whakawehe 1 ki te -2.

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

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.

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

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

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

Tāpiri -\frac{1}{2} ki te \frac{1}{16} 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.

$$x^{2}+\frac{1}{2}x+\frac{1}{16}=-\frac{7}{16}$$

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}.

$$\left(x+\frac{1}{4}\right)^{2}=-\frac{7}{16}$$

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

$$\sqrt{\left(x+\frac{1}{4}\right)^{2}}=\sqrt{-\frac{7}{16}}$$

Whakarūnātia.

$$x+\frac{1}{4}=\frac{\sqrt{7}i}{4} x+\frac{1}{4}=-\frac{\sqrt{7}i}{4}$$

Me tango \frac{1}{4} mai i ngā taha e rua o te whārite.

$$x=\frac{-1+\sqrt{7}i}{4} x=\frac{-\sqrt{7}i-1}{4}$$