Solve 24=(x+4)^2

$$24 = ( x + 4 ) ^ { 2 }$$

Solve for x

$$x=2\sqrt{6}-4\approx 0.898979486$$

$$x=-2\sqrt{6}-4\approx -8.898979486$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+4\right)^{2}.

$$24=x^{2}+8x+16$$

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

$$x^{2}+8x+16=24$$

Subtract 24 from both sides.

$$x^{2}+8x+16-24=0$$

Subtract 24 from 16 to get -8.

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

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 8 for b, and -8 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

Square 8.

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

Multiply -4 times -8.

$$x=\frac{-8±\sqrt{64+32}}{2}$$

Add 64 to 32.

$$x=\frac{-8±\sqrt{96}}{2}$$

Take the square root of 96.

$$x=\frac{-8±4\sqrt{6}}{2}$$

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

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

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

$$x=2\sqrt{6}-4$$

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

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

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

$$x=-2\sqrt{6}-4$$

The equation is now solved.

$$x=2\sqrt{6}-4 x=-2\sqrt{6}-4$$

Steps for Completing the Square

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+4\right)^{2}.

$$24=x^{2}+8x+16$$

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

$$x^{2}+8x+16=24$$

Factor x^{2}+8x+16. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

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

Take the square root of both sides of the equation.

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

Simplify.

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

Subtract 4 from both sides of the equation.

$$x=2\sqrt{6}-4 x=-2\sqrt{6}-4$$