Solve left(x+6right)^2=4

$${ \left(x+6 \right) }^{ 2 } =4$$

Solve for x

$$x=-4$$

$$x=-8$$

Steps Using Factoring

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+6\right)^{2}.

$$x^{2}+12x+36=4$$

Subtract 4 from both sides.

$$x^{2}+12x+36-4=0$$

Subtract 4 from 36 to get 32.

$$x^{2}+12x+32=0$$

To solve the equation, factor x^{2}+12x+32 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.

$$a+b=12 ab=32$$

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 32.

$$1,32 2,16 4,8$$

Calculate the sum for each pair.

$$1+32=33 2+16=18 4+8=12$$

The solution is the pair that gives sum 12.

$$a=4 b=8$$

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

$$\left(x+4\right)\left(x+8\right)$$

To find equation solutions, solve x+4=0 and x+8=0.

$$x=-4 x=-8$$

Steps Using Factoring By Grouping

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+6\right)^{2}.

$$x^{2}+12x+36=4$$

Subtract 4 from both sides.

$$x^{2}+12x+36-4=0$$

Subtract 4 from 36 to get 32.

$$x^{2}+12x+32=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx+32. To find a and b, set up a system to be solved.

$$a+b=12 ab=1\times 32=32$$

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 32.

$$1,32 2,16 4,8$$

Calculate the sum for each pair.

$$1+32=33 2+16=18 4+8=12$$

The solution is the pair that gives sum 12.

$$a=4 b=8$$

Rewrite x^{2}+12x+32 as \left(x^{2}+4x\right)+\left(8x+32\right).

$$\left(x^{2}+4x\right)+\left(8x+32\right)$$

Factor out x in the first and 8 in the second group.

$$x\left(x+4\right)+8\left(x+4\right)$$

Factor out common term x+4 by using distributive property.

$$\left(x+4\right)\left(x+8\right)$$

To find equation solutions, solve x+4=0 and x+8=0.

$$x=-4 x=-8$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+6\right)^{2}.

$$x^{2}+12x+36=4$$

Subtract 4 from both sides.

$$x^{2}+12x+36-4=0$$

Subtract 4 from 36 to get 32.

$$x^{2}+12x+32=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 12 for b, and 32 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-12±\sqrt{12^{2}-4\times 32}}{2}$$

Square 12.

$$x=\frac{-12±\sqrt{144-4\times 32}}{2}$$

Multiply -4 times 32.

$$x=\frac{-12±\sqrt{144-128}}{2}$$

Add 144 to -128.

$$x=\frac{-12±\sqrt{16}}{2}$$

Take the square root of 16.

$$x=\frac{-12±4}{2}$$

Now solve the equation x=\frac{-12±4}{2} when ± is plus. Add -12 to 4.

$$x=-\frac{8}{2}$$

Divide -8 by 2.

$$x=-4$$

Now solve the equation x=\frac{-12±4}{2} when ± is minus. Subtract 4 from -12.

$$x=-\frac{16}{2}$$

Divide -16 by 2.

$$x=-8$$

The equation is now solved.

$$x=-4 x=-8$$

Steps for Completing the Square

Take the square root of both sides of the equation.

$$\sqrt{\left(x+6\right)^{2}}=\sqrt{4}$$

Simplify.

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

Subtract 6 from both sides of the equation.

$$x=-4 x=-8$$