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

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

Whakaoti mō x (complex solution)

$$x=\frac{-\sqrt{31}i-1}{2}\approx -0.5-2.783882181i$$

$$x=\frac{-1+\sqrt{31}i}{2}\approx -0.5+2.783882181i$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tangohia te x^{2} mai i ngā taha e rua.

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

Tangohia te 3x mai i ngā taha e rua.

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

Pahekotia te 2x me -3x, ka -x.

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

Tangohia te 5 mai i ngā taha e rua.

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

Tangohia te 5 i te -3, ka -8.

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

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.

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

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

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

Whakareatia -4 ki te -1.

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

Whakareatia 4 ki te -8.

$$x=\frac{-\left(-1\right)±\sqrt{1-32}}{2\left(-1\right)}$$

Tāpiri 1 ki te -32.

$$x=\frac{-\left(-1\right)±\sqrt{-31}}{2\left(-1\right)}$$

Tuhia te pūtakerua o te -31.

$$x=\frac{-\left(-1\right)±\sqrt{31}i}{2\left(-1\right)}$$

Ko te tauaro o -1 ko 1.

$$x=\frac{1±\sqrt{31}i}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$x=\frac{1±\sqrt{31}i}{-2}$$

Nā, me whakaoti te whārite x=\frac{1±\sqrt{31}i}{-2} ina he tāpiri te ±. Tāpiri 1 ki te i\sqrt{31}.

$$x=\frac{1+\sqrt{31}i}{-2}$$

Whakawehe 1+i\sqrt{31} ki te -2.

$$x=\frac{-\sqrt{31}i-1}{2}$$

Nā, me whakaoti te whārite x=\frac{1±\sqrt{31}i}{-2} ina he tango te ±. Tango i\sqrt{31} mai i 1.

$$x=\frac{-\sqrt{31}i+1}{-2}$$

Whakawehe 1-i\sqrt{31} ki te -2.

$$x=\frac{-1+\sqrt{31}i}{2}$$

Kua oti te whārite te whakatau.

$$x=\frac{-\sqrt{31}i-1}{2} x=\frac{-1+\sqrt{31}i}{2}$$

Ngā Upane mō te Whakaoti i te Pūrua

Tangohia te x^{2} mai i ngā taha e rua.

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

Tangohia te 3x mai i ngā taha e rua.

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

Pahekotia te 2x me -3x, ka -x.

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

Me tāpiri te 3 ki ngā taha e rua.

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

Tāpirihia te 5 ki te 3, ka 8.

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

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.

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

Whakawehea ngā taha e rua ki te -1.

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

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

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

Whakawehe -1 ki te -1.

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

Whakawehe 8 ki te -1.

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

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.

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

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

$$x^{2}+x+\frac{1}{4}=-8+\frac{1}{4}$$

Tāpiri -8 ki te \frac{1}{4}.

$$x^{2}+x+\frac{1}{4}=-\frac{31}{4}$$

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

$$\left(x+\frac{1}{2}\right)^{2}=-\frac{31}{4}$$

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

$$\sqrt{\left(x+\frac{1}{2}\right)^{2}}=\sqrt{-\frac{31}{4}}$$

Whakarūnātia.

$$x+\frac{1}{2}=\frac{\sqrt{31}i}{2} x+\frac{1}{2}=-\frac{\sqrt{31}i}{2}$$

Me tango \frac{1}{2} mai i ngā taha e rua o te whārite.

$$x=\frac{-1+\sqrt{31}i}{2} x=\frac{-\sqrt{31}i-1}{2}$$