Whakaoti i te x^2+(x+2)^2+4(x+2)-1=0

$$x ^ { 2 } + ( x + 2 ) ^ { 2 } + 4 ( x + 2 ) - 1 = 0$$

Whakaoti mō x (complex solution)

$$x=\frac{\sqrt{6}i}{2}-2\approx -2+1.224744871i$$

$$x=-\frac{\sqrt{6}i}{2}-2\approx -2-1.224744871i$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(x+2\right)^{2}.

$$x^{2}+x^{2}+4x+4+4\left(x+2\right)-1=0$$

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

$$2x^{2}+4x+4+4\left(x+2\right)-1=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te 4 ki te x+2.

$$2x^{2}+4x+4+4x+8-1=0$$

Pahekotia te 4x me 4x, ka 8x.

$$2x^{2}+8x+4+8-1=0$$

Tāpirihia te 4 ki te 8, ka 12.

$$2x^{2}+8x+12-1=0$$

Tangohia te 1 i te 12, ka 11.

$$2x^{2}+8x+11=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 2 mō a, 8 mō b, me 11 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-8±\sqrt{8^{2}-4\times 2\times 11}}{2\times 2}$$

Pūrua 8.

$$x=\frac{-8±\sqrt{64-4\times 2\times 11}}{2\times 2}$$

Whakareatia -4 ki te 2.

$$x=\frac{-8±\sqrt{64-8\times 11}}{2\times 2}$$

Whakareatia -8 ki te 11.

$$x=\frac{-8±\sqrt{64-88}}{2\times 2}$$

Tāpiri 64 ki te -88.

$$x=\frac{-8±\sqrt{-24}}{2\times 2}$$

Tuhia te pūtakerua o te -24.

$$x=\frac{-8±2\sqrt{6}i}{2\times 2}$$

Whakareatia 2 ki te 2.

$$x=\frac{-8±2\sqrt{6}i}{4}$$

Nā, me whakaoti te whārite x=\frac{-8±2\sqrt{6}i}{4} ina he tāpiri te ±. Tāpiri -8 ki te 2i\sqrt{6}.

$$x=\frac{-8+2\sqrt{6}i}{4}$$

Whakawehe -8+2i\sqrt{6} ki te 4.

$$x=\frac{\sqrt{6}i}{2}-2$$

Nā, me whakaoti te whārite x=\frac{-8±2\sqrt{6}i}{4} ina he tango te ±. Tango 2i\sqrt{6} mai i -8.

$$x=\frac{-2\sqrt{6}i-8}{4}$$

Whakawehe -8-2i\sqrt{6} ki te 4.

$$x=-\frac{\sqrt{6}i}{2}-2$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{6}i}{2}-2 x=-\frac{\sqrt{6}i}{2}-2$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(x+2\right)^{2}.

$$x^{2}+x^{2}+4x+4+4\left(x+2\right)-1=0$$

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

$$2x^{2}+4x+4+4\left(x+2\right)-1=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te 4 ki te x+2.

$$2x^{2}+4x+4+4x+8-1=0$$

Pahekotia te 4x me 4x, ka 8x.

$$2x^{2}+8x+4+8-1=0$$

Tāpirihia te 4 ki te 8, ka 12.

$$2x^{2}+8x+12-1=0$$

Tangohia te 1 i te 12, ka 11.

$$2x^{2}+8x+11=0$$

Tangohia te 11 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$2x^{2}+8x=-11$$

Whakawehea ngā taha e rua ki te 2.

$$\frac{2x^{2}+8x}{2}=-\frac{11}{2}$$

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

$$x^{2}+\frac{8}{2}x=-\frac{11}{2}$$

Whakawehe 8 ki te 2.

$$x^{2}+4x=-\frac{11}{2}$$

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.

$$x^{2}+4x+2^{2}=-\frac{11}{2}+2^{2}$$

Pūrua 2.

$$x^{2}+4x+4=-\frac{11}{2}+4$$

Tāpiri -\frac{11}{2} ki te 4.

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

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}.

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

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

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

Whakarūnātia.

$$x+2=\frac{\sqrt{6}i}{2} x+2=-\frac{\sqrt{6}i}{2}$$

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

$$x=\frac{\sqrt{6}i}{2}-2 x=-\frac{\sqrt{6}i}{2}-2$$