Solve (x+4)^2=4

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

Solve for x

$$x=-2$$

$$x=-6$$

Steps Using Factoring

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

$$x^{2}+8x+16=4$$

Subtract 4 from both sides.

$$x^{2}+8x+16-4=0$$

Subtract 4 from 16 to get 12.

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

To solve the equation, factor x^{2}+8x+12 using formula x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). To find a and b, set up a system to be solved.

$$a+b=8 ab=12$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 12.

$$1,12 2,6 3,4$$

Calculate the sum for each pair.

$$1+12=13 2+6=8 3+4=7$$

The solution is the pair that gives sum 8.

$$a=2 b=6$$

Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.

$$\left(x+2\right)\left(x+6\right)$$

To find equation solutions, solve x+2=0 and x+6=0.

$$x=-2 x=-6$$

Steps Using Factoring By Grouping

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

$$x^{2}+8x+16=4$$

Subtract 4 from both sides.

$$x^{2}+8x+16-4=0$$

Subtract 4 from 16 to get 12.

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

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx+12. To find a and b, set up a system to be solved.

$$a+b=8 ab=1\times 12=12$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 12.

$$1,12 2,6 3,4$$

Calculate the sum for each pair.

$$1+12=13 2+6=8 3+4=7$$

The solution is the pair that gives sum 8.

$$a=2 b=6$$

Rewrite x^{2}+8x+12 as \left(x^{2}+2x\right)+\left(6x+12\right).

$$\left(x^{2}+2x\right)+\left(6x+12\right)$$

Factor out x in the first and 6 in the second group.

$$x\left(x+2\right)+6\left(x+2\right)$$

Factor out common term x+2 by using distributive property.

$$\left(x+2\right)\left(x+6\right)$$

To find equation solutions, solve x+2=0 and x+6=0.

$$x=-2 x=-6$$

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}.

$$x^{2}+8x+16=4$$

Subtract 4 from both sides.

$$x^{2}+8x+16-4=0$$

Subtract 4 from 16 to get 12.

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

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 8 for b, and 12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-8±\sqrt{8^{2}-4\times 12}}{2}$$

Square 8.

$$x=\frac{-8±\sqrt{64-4\times 12}}{2}$$

Multiply -4 times 12.

$$x=\frac{-8±\sqrt{64-48}}{2}$$

Add 64 to -48.

$$x=\frac{-8±\sqrt{16}}{2}$$

Take the square root of 16.

$$x=\frac{-8±4}{2}$$

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

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

Divide -4 by 2.

$$x=-2$$

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

$$x=-\frac{12}{2}$$

Divide -12 by 2.

$$x=-6$$

The equation is now solved.

$$x=-2 x=-6$$

Steps for Completing the Square

Take the square root of both sides of the equation.

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

Simplify.

$$x+4=2 x+4=-2$$

Subtract 4 from both sides of the equation.

$$x=-2 x=-6$$