Solve x^2+(x+2)^2+4(x+2)-1=0

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

Solve for x (complex solution)

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

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

Steps Using the Quadratic Formula

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+2\right)^{2}.

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

Combine x^{2} and x^{2} to get 2x^{2}.

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

Use the distributive property to multiply 4 by x+2.

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

Combine 4x and 4x to get 8x.

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

Add 4 and 8 to get 12.

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

Subtract 1 from 12 to get 11.

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

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 8 for b, and 11 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

Square 8.

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

Multiply -4 times 2.

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

Multiply -8 times 11.

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

Add 64 to -88.

$$x=\frac{-8±\sqrt{-24}}{2\times 2}$$

Take the square root of -24.

$$x=\frac{-8±2\sqrt{6}i}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-8±2\sqrt{6}i}{4}$$

Now solve the equation x=\frac{-8±2\sqrt{6}i}{4} when ± is plus. Add -8 to 2i\sqrt{6}.

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

Divide -8+2i\sqrt{6} by 4.

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

Now solve the equation x=\frac{-8±2\sqrt{6}i}{4} when ± is minus. Subtract 2i\sqrt{6} from -8.

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

Divide -8-2i\sqrt{6} by 4.

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

The equation is now solved.

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

Steps for Completing the Square

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+2\right)^{2}.

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

Combine x^{2} and x^{2} to get 2x^{2}.

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

Use the distributive property to multiply 4 by x+2.

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

Combine 4x and 4x to get 8x.

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

Add 4 and 8 to get 12.

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

Subtract 1 from 12 to get 11.

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

Subtract 11 from both sides. Anything subtracted from zero gives its negation.

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

Divide both sides by 2.

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

Dividing by 2 undoes the multiplication by 2.

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

Divide 8 by 2.

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

Divide 4, the coefficient of the x term, by 2 to get 2. Then add the square of 2 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square 2.

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

Add -\frac{11}{2} to 4.

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

Factor x^{2}+4x+4. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

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

Take the square root of both sides of the equation.

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

Simplify.

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

Subtract 2 from both sides of the equation.

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