Whakaoti i te 8x+2-3x^2+1=35-x^2

$$8 x + 2 - 3 x ^ { 2 } + 1 = 35 - x ^ { 2 }$$

Whakaoti mō x (complex solution)

$$x=2+2\sqrt{3}i\approx 2+3.464101615i$$

$$x=-2\sqrt{3}i+2\approx 2-3.464101615i$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tāpirihia te 2 ki te 1, ka 3.

$$8x+3-3x^{2}=35-x^{2}$$

Tangohia te 35 mai i ngā taha e rua.

$$8x+3-3x^{2}-35=-x^{2}$$

Tangohia te 35 i te 3, ka -32.

$$8x-32-3x^{2}=-x^{2}$$

Me tāpiri te x^{2} ki ngā taha e rua.

$$8x-32-3x^{2}+x^{2}=0$$

Pahekotia te -3x^{2} me x^{2}, ka -2x^{2}.

$$8x-32-2x^{2}=0$$

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.

$$-2x^{2}+8x-32=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -2 mō a, 8 mō b, me -32 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-8±\sqrt{8^{2}-4\left(-2\right)\left(-32\right)}}{2\left(-2\right)}$$

Pūrua 8.

$$x=\frac{-8±\sqrt{64-4\left(-2\right)\left(-32\right)}}{2\left(-2\right)}$$

Whakareatia -4 ki te -2.

$$x=\frac{-8±\sqrt{64+8\left(-32\right)}}{2\left(-2\right)}$$

Whakareatia 8 ki te -32.

$$x=\frac{-8±\sqrt{64-256}}{2\left(-2\right)}$$

Tāpiri 64 ki te -256.

$$x=\frac{-8±\sqrt{-192}}{2\left(-2\right)}$$

Tuhia te pūtakerua o te -192.

$$x=\frac{-8±8\sqrt{3}i}{2\left(-2\right)}$$

Whakareatia 2 ki te -2.

$$x=\frac{-8±8\sqrt{3}i}{-4}$$

Nā, me whakaoti te whārite x=\frac{-8±8\sqrt{3}i}{-4} ina he tāpiri te ±. Tāpiri -8 ki te 8i\sqrt{3}.

$$x=\frac{-8+8\sqrt{3}i}{-4}$$

Whakawehe -8+8i\sqrt{3} ki te -4.

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

Nā, me whakaoti te whārite x=\frac{-8±8\sqrt{3}i}{-4} ina he tango te ±. Tango 8i\sqrt{3} mai i -8.

$$x=\frac{-8\sqrt{3}i-8}{-4}$$

Whakawehe -8-8i\sqrt{3} ki te -4.

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

Kua oti te whārite te whakatau.

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

Ngā Upane mō te Whakaoti i te Pūrua

Tāpirihia te 2 ki te 1, ka 3.

$$8x+3-3x^{2}=35-x^{2}$$

Me tāpiri te x^{2} ki ngā taha e rua.

$$8x+3-3x^{2}+x^{2}=35$$

Pahekotia te -3x^{2} me x^{2}, ka -2x^{2}.

$$8x+3-2x^{2}=35$$

Tangohia te 3 mai i ngā taha e rua.

$$8x-2x^{2}=35-3$$

Tangohia te 3 i te 35, ka 32.

$$8x-2x^{2}=32$$

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.

$$-2x^{2}+8x=32$$

Whakawehea ngā taha e rua ki te -2.

$$\frac{-2x^{2}+8x}{-2}=\frac{32}{-2}$$

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

$$x^{2}+\frac{8}{-2}x=\frac{32}{-2}$$

Whakawehe 8 ki te -2.

$$x^{2}-4x=\frac{32}{-2}$$

Whakawehe 32 ki te -2.

$$x^{2}-4x=-16$$

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.

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

Pūrua -2.

$$x^{2}-4x+4=-16+4$$

Tāpiri -16 ki te 4.

$$x^{2}-4x+4=-12$$

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}.

$$\left(x-2\right)^{2}=-12$$

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

$$\sqrt{\left(x-2\right)^{2}}=\sqrt{-12}$$

Whakarūnātia.

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

Me tāpiri 2 ki ngā taha e rua o te whārite.

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