Solve (x+9)^2=4

$$( x + 9 ) ^ { 2 } = 4$$

Solve for x

$$x=-7$$

$$x=-11$$

Steps Using Factoring

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+9\right)^{2}.

$$x^{2}+18x+81=4$$

Subtract 4 from both sides.

$$x^{2}+18x+81-4=0$$

Subtract 4 from 81 to get 77.

$$x^{2}+18x+77=0$$

To solve the equation, factor x^{2}+18x+77 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.

$$a+b=18 ab=77$$

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 77.

$$1,77 7,11$$

Calculate the sum for each pair.

$$1+77=78 7+11=18$$

The solution is the pair that gives sum 18.

$$a=7 b=11$$

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

$$\left(x+7\right)\left(x+11\right)$$

To find equation solutions, solve x+7=0 and x+11=0.

$$x=-7 x=-11$$

Steps Using Factoring By Grouping

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+9\right)^{2}.

$$x^{2}+18x+81=4$$

Subtract 4 from both sides.

$$x^{2}+18x+81-4=0$$

Subtract 4 from 81 to get 77.

$$x^{2}+18x+77=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx+77. To find a and b, set up a system to be solved.

$$a+b=18 ab=1\times 77=77$$

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 77.

$$1,77 7,11$$

Calculate the sum for each pair.

$$1+77=78 7+11=18$$

The solution is the pair that gives sum 18.

$$a=7 b=11$$

Rewrite x^{2}+18x+77 as \left(x^{2}+7x\right)+\left(11x+77\right).

$$\left(x^{2}+7x\right)+\left(11x+77\right)$$

Factor out x in the first and 11 in the second group.

$$x\left(x+7\right)+11\left(x+7\right)$$

Factor out common term x+7 by using distributive property.

$$\left(x+7\right)\left(x+11\right)$$

To find equation solutions, solve x+7=0 and x+11=0.

$$x=-7 x=-11$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+9\right)^{2}.

$$x^{2}+18x+81=4$$

Subtract 4 from both sides.

$$x^{2}+18x+81-4=0$$

Subtract 4 from 81 to get 77.

$$x^{2}+18x+77=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 18 for b, and 77 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-18±\sqrt{18^{2}-4\times 77}}{2}$$

Square 18.

$$x=\frac{-18±\sqrt{324-4\times 77}}{2}$$

Multiply -4 times 77.

$$x=\frac{-18±\sqrt{324-308}}{2}$$

Add 324 to -308.

$$x=\frac{-18±\sqrt{16}}{2}$$

Take the square root of 16.

$$x=\frac{-18±4}{2}$$

Now solve the equation x=\frac{-18±4}{2} when ± is plus. Add -18 to 4.

$$x=-\frac{14}{2}$$

Divide -14 by 2.

$$x=-7$$

Now solve the equation x=\frac{-18±4}{2} when ± is minus. Subtract 4 from -18.

$$x=-\frac{22}{2}$$

Divide -22 by 2.

$$x=-11$$

The equation is now solved.

$$x=-7 x=-11$$

Steps for Completing the Square

Take the square root of both sides of the equation.

$$\sqrt{\left(x+9\right)^{2}}=\sqrt{4}$$

Simplify.

$$x+9=2 x+9=-2$$

Subtract 9 from both sides of the equation.

$$x=-7 x=-11$$