Whakaoti i te x-2x^2=8

$$x - 2 x ^ { 2 } = 8$$

Whakaoti mō x (complex solution)

$$x=\frac{-3\sqrt{7}i+1}{4}\approx 0.25-1.984313483i$$

$$x=\frac{1+3\sqrt{7}i}{4}\approx 0.25+1.984313483i$$

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.

$$-2x^{2}+x=8$$

Me tango 8 mai i ngā taha e rua o te whārite.

$$-2x^{2}+x-8=8-8$$

Mā te tango i te 8 i a ia ake anō ka toe ko te 0.

$$-2x^{2}+x-8=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -2 mō a, 1 mō b, me -8 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

Pūrua 1.

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

Whakareatia -4 ki te -2.

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

Whakareatia 8 ki te -8.

$$x=\frac{-1±\sqrt{1-64}}{2\left(-2\right)}$$

Tāpiri 1 ki te -64.

$$x=\frac{-1±\sqrt{-63}}{2\left(-2\right)}$$

Tuhia te pūtakerua o te -63.

$$x=\frac{-1±3\sqrt{7}i}{2\left(-2\right)}$$

Whakareatia 2 ki te -2.

$$x=\frac{-1±3\sqrt{7}i}{-4}$$

Nā, me whakaoti te whārite x=\frac{-1±3\sqrt{7}i}{-4} ina he tāpiri te ±. Tāpiri -1 ki te 3i\sqrt{7}.

$$x=\frac{-1+3\sqrt{7}i}{-4}$$

Whakawehe -1+3i\sqrt{7} ki te -4.

$$x=\frac{-3\sqrt{7}i+1}{4}$$

Nā, me whakaoti te whārite x=\frac{-1±3\sqrt{7}i}{-4} ina he tango te ±. Tango 3i\sqrt{7} mai i -1.

$$x=\frac{-3\sqrt{7}i-1}{-4}$$

Whakawehe -1-3i\sqrt{7} ki te -4.

$$x=\frac{1+3\sqrt{7}i}{4}$$

Kua oti te whārite te whakatau.

$$x=\frac{-3\sqrt{7}i+1}{4} x=\frac{1+3\sqrt{7}i}{4}$$

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.

$$-2x^{2}+x=8$$

Whakawehea ngā taha e rua ki te -2.

$$\frac{-2x^{2}+x}{-2}=\frac{8}{-2}$$

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

$$x^{2}+\frac{1}{-2}x=\frac{8}{-2}$$

Whakawehe 1 ki te -2.

$$x^{2}-\frac{1}{2}x=\frac{8}{-2}$$

Whakawehe 8 ki te -2.

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

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.

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

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

$$x^{2}-\frac{1}{2}x+\frac{1}{16}=-4+\frac{1}{16}$$

Tāpiri -4 ki te \frac{1}{16}.

$$x^{2}-\frac{1}{2}x+\frac{1}{16}=-\frac{63}{16}$$

Tauwehea te x^{2}-\frac{1}{2}x+\frac{1}{16}. Ko te tikanga, ina ko x^{2}+bx+c he pūrua tika, ka taea te tauwehe i ngā wā katoa hei \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1}{4}\right)^{2}=-\frac{63}{16}$$

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

$$\sqrt{\left(x-\frac{1}{4}\right)^{2}}=\sqrt{-\frac{63}{16}}$$

Whakarūnātia.

$$x-\frac{1}{4}=\frac{3\sqrt{7}i}{4} x-\frac{1}{4}=-\frac{3\sqrt{7}i}{4}$$

Me tāpiri \frac{1}{4} ki ngā taha e rua o te whārite.

$$x=\frac{1+3\sqrt{7}i}{4} x=\frac{-3\sqrt{7}i+1}{4}$$