Whakaoti i te 5-x^2+3x=12

$$5- { x }^{ 2 } +3x=12$$

Whakaoti mō x (complex solution)

$$x=\frac{-\sqrt{19}i+3}{2}\approx 1.5-2.179449472i$$

$$x=\frac{3+\sqrt{19}i}{2}\approx 1.5+2.179449472i$$

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.

$$-x^{2}+3x+5=12$$

Me tango 12 mai i ngā taha e rua o te whārite.

$$-x^{2}+3x+5-12=12-12$$

Mā te tango i te 12 i a ia ake anō ka toe ko te 0.

$$-x^{2}+3x+5-12=0$$

Tango 12 mai i 5.

$$-x^{2}+3x-7=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -1 mō a, 3 mō b, me -7 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-3±\sqrt{3^{2}-4\left(-1\right)\left(-7\right)}}{2\left(-1\right)}$$

Pūrua 3.

$$x=\frac{-3±\sqrt{9-4\left(-1\right)\left(-7\right)}}{2\left(-1\right)}$$

Whakareatia -4 ki te -1.

$$x=\frac{-3±\sqrt{9+4\left(-7\right)}}{2\left(-1\right)}$$

Whakareatia 4 ki te -7.

$$x=\frac{-3±\sqrt{9-28}}{2\left(-1\right)}$$

Tāpiri 9 ki te -28.

$$x=\frac{-3±\sqrt{-19}}{2\left(-1\right)}$$

Tuhia te pūtakerua o te -19.

$$x=\frac{-3±\sqrt{19}i}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$x=\frac{-3±\sqrt{19}i}{-2}$$

Nā, me whakaoti te whārite x=\frac{-3±\sqrt{19}i}{-2} ina he tāpiri te ±. Tāpiri -3 ki te i\sqrt{19}.

$$x=\frac{-3+\sqrt{19}i}{-2}$$

Whakawehe -3+i\sqrt{19} ki te -2.

$$x=\frac{-\sqrt{19}i+3}{2}$$

Nā, me whakaoti te whārite x=\frac{-3±\sqrt{19}i}{-2} ina he tango te ±. Tango i\sqrt{19} mai i -3.

$$x=\frac{-\sqrt{19}i-3}{-2}$$

Whakawehe -3-i\sqrt{19} ki te -2.

$$x=\frac{3+\sqrt{19}i}{2}$$

Kua oti te whārite te whakatau.

$$x=\frac{-\sqrt{19}i+3}{2} x=\frac{3+\sqrt{19}i}{2}$$

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.

$$-x^{2}+3x+5=12$$

Me tango 5 mai i ngā taha e rua o te whārite.

$$-x^{2}+3x+5-5=12-5$$

Mā te tango i te 5 i a ia ake anō ka toe ko te 0.

$$-x^{2}+3x=12-5$$

Tango 5 mai i 12.

$$-x^{2}+3x=7$$

Whakawehea ngā taha e rua ki te -1.

$$\frac{-x^{2}+3x}{-1}=\frac{7}{-1}$$

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

$$x^{2}+\frac{3}{-1}x=\frac{7}{-1}$$

Whakawehe 3 ki te -1.

$$x^{2}-3x=\frac{7}{-1}$$

Whakawehe 7 ki te -1.

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

Whakawehea te -3, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{3}{2}. Nā, tāpiria te pūrua o te -\frac{3}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

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

Pūruatia -\frac{3}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-3x+\frac{9}{4}=-7+\frac{9}{4}$$

Tāpiri -7 ki te \frac{9}{4}.

$$x^{2}-3x+\frac{9}{4}=-\frac{19}{4}$$

Tauwehea te x^{2}-3x+\frac{9}{4}. 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}.

$$\left(x-\frac{3}{2}\right)^{2}=-\frac{19}{4}$$

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

$$\sqrt{\left(x-\frac{3}{2}\right)^{2}}=\sqrt{-\frac{19}{4}}$$

Whakarūnātia.

$$x-\frac{3}{2}=\frac{\sqrt{19}i}{2} x-\frac{3}{2}=-\frac{\sqrt{19}i}{2}$$

Me tāpiri \frac{3}{2} ki ngā taha e rua o te whārite.

$$x=\frac{3+\sqrt{19}i}{2} x=\frac{-\sqrt{19}i+3}{2}$$