Solve 0=2(x+2)^2+24

$$0 = 2 ( x + 2 ) ^ { 2 } + 24$$

Solve for x (complex solution)

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

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

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}.

$$0=2\left(x^{2}+4x+4\right)+24$$

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

$$0=2x^{2}+8x+8+24$$

Add 8 and 24 to get 32.

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

Swap sides so that all variable terms are on the left hand side.

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

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 8 for b, and 32 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-8±\sqrt{8^{2}-4\times 2\times 32}}{2\times 2}$$

Square 8.

$$x=\frac{-8±\sqrt{64-4\times 2\times 32}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-8±\sqrt{64-8\times 32}}{2\times 2}$$

Multiply -8 times 32.

$$x=\frac{-8±\sqrt{64-256}}{2\times 2}$$

Add 64 to -256.

$$x=\frac{-8±\sqrt{-192}}{2\times 2}$$

Take the square root of -192.

$$x=\frac{-8±8\sqrt{3}i}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-8±8\sqrt{3}i}{4}$$

Now solve the equation x=\frac{-8±8\sqrt{3}i}{4} when ± is plus. Add -8 to 8i\sqrt{3}.

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

Divide -8+8i\sqrt{3} by 4.

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

Now solve the equation x=\frac{-8±8\sqrt{3}i}{4} when ± is minus. Subtract 8i\sqrt{3} from -8.

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

Divide -8-8i\sqrt{3} by 4.

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

The equation is now solved.

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

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}.

$$0=2\left(x^{2}+4x+4\right)+24$$

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

$$0=2x^{2}+8x+8+24$$

Add 8 and 24 to get 32.

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

Swap sides so that all variable terms are on the left hand side.

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

Subtract 32 from both sides. Anything subtracted from zero gives its negation.

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

Divide both sides by 2.

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

Dividing by 2 undoes the multiplication by 2.

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

Divide 8 by 2.

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

Divide -32 by 2.

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

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.

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

Square 2.

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

Add -16 to 4.

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

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}.

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

Take the square root of both sides of the equation.

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

Simplify.

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

Subtract 2 from both sides of the equation.

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