Whakaoti i te (-4x-3)-x^2-3=0

$$(-4x-3)- { x }^{ 2 } -3=0$$

Whakaoti mō x (complex solution)

$$x=-\sqrt{2}i-2\approx -2-1.414213562i$$

$$x=-2+\sqrt{2}i\approx -2+1.414213562i$$

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.

$$-x^{2}-4x-6=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -1 mō a, -4 mō b, me -6 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\left(-1\right)\left(-6\right)}}{2\left(-1\right)}$$

Pūrua -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\left(-1\right)\left(-6\right)}}{2\left(-1\right)}$$

Whakareatia -4 ki te -1.

$$x=\frac{-\left(-4\right)±\sqrt{16+4\left(-6\right)}}{2\left(-1\right)}$$

Whakareatia 4 ki te -6.

$$x=\frac{-\left(-4\right)±\sqrt{16-24}}{2\left(-1\right)}$$

Tāpiri 16 ki te -24.

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

Tuhia te pūtakerua o te -8.

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

Ko te tauaro o -4 ko 4.

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

Whakareatia 2 ki te -1.

$$x=\frac{4±2\sqrt{2}i}{-2}$$

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

$$x=\frac{4+2\sqrt{2}i}{-2}$$

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

$$x=-\sqrt{2}i-2$$

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

$$x=\frac{-2\sqrt{2}i+4}{-2}$$

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

$$x=-2+\sqrt{2}i$$

Kua oti te whārite te whakatau.

$$x=-\sqrt{2}i-2 x=-2+\sqrt{2}i$$

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.

$$-x^{2}-4x-6=0$$

Me tāpiri 6 ki ngā taha e rua o te whārite.

$$-x^{2}-4x-6-\left(-6\right)=-\left(-6\right)$$

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

$$-x^{2}-4x=-\left(-6\right)$$

Tango -6 mai i 0.

$$-x^{2}-4x=6$$

Whakawehea ngā taha e rua ki te -1.

$$\frac{-x^{2}-4x}{-1}=\frac{6}{-1}$$

Mā te whakawehe ki te -1 ka wetekia te whakareanga ki te -1.

$$x^{2}+\left(-\frac{4}{-1}\right)x=\frac{6}{-1}$$

Whakawehe -4 ki te -1.

$$x^{2}+4x=\frac{6}{-1}$$

Whakawehe 6 ki te -1.

$$x^{2}+4x=-6$$

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

$$x^{2}+4x+2^{2}=-6+2^{2}$$

Pūrua 2.

$$x^{2}+4x+4=-6+4$$

Tāpiri -6 ki te 4.

$$x^{2}+4x+4=-2$$

Tauwehea x^{2}+4x+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}.

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

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

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

Whakarūnātia.

$$x+2=\sqrt{2}i x+2=-\sqrt{2}i$$

Me tango 2 mai i ngā taha e rua o te whārite.

$$x=-2+\sqrt{2}i x=-\sqrt{2}i-2$$