Solve (x+9)^2=36

$$( x + 9 ) ^ { 2 } = 36$$

Solve for x

$$x=-3$$

$$x=-15$$

Steps Using Factoring

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

$$x^{2}+18x+81=36$$

Subtract 36 from both sides.

$$x^{2}+18x+81-36=0$$

Subtract 36 from 81 to get 45.

$$x^{2}+18x+45=0$$

To solve the equation, factor x^{2}+18x+45 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.

$$a+b=18 ab=45$$

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 45.

$$1,45 3,15 5,9$$

Calculate the sum for each pair.

$$1+45=46 3+15=18 5+9=14$$

The solution is the pair that gives sum 18.

$$a=3 b=15$$

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

$$\left(x+3\right)\left(x+15\right)$$

To find equation solutions, solve x+3=0 and x+15=0.

$$x=-3 x=-15$$

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}.

$$x^{2}+18x+81=36$$

Subtract 36 from both sides.

$$x^{2}+18x+81-36=0$$

Subtract 36 from 81 to get 45.

$$x^{2}+18x+45=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx+45. To find a and b, set up a system to be solved.

$$a+b=18 ab=1\times 45=45$$

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 45.

$$1,45 3,15 5,9$$

Calculate the sum for each pair.

$$1+45=46 3+15=18 5+9=14$$

The solution is the pair that gives sum 18.

$$a=3 b=15$$

Rewrite x^{2}+18x+45 as \left(x^{2}+3x\right)+\left(15x+45\right).

$$\left(x^{2}+3x\right)+\left(15x+45\right)$$

Factor out x in the first and 15 in the second group.

$$x\left(x+3\right)+15\left(x+3\right)$$

Factor out common term x+3 by using distributive property.

$$\left(x+3\right)\left(x+15\right)$$

To find equation solutions, solve x+3=0 and x+15=0.

$$x=-3 x=-15$$

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}.

$$x^{2}+18x+81=36$$

Subtract 36 from both sides.

$$x^{2}+18x+81-36=0$$

Subtract 36 from 81 to get 45.

$$x^{2}+18x+45=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 18 for b, and 45 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-18±\sqrt{18^{2}-4\times 45}}{2}$$

Square 18.

$$x=\frac{-18±\sqrt{324-4\times 45}}{2}$$

Multiply -4 times 45.

$$x=\frac{-18±\sqrt{324-180}}{2}$$

Add 324 to -180.

$$x=\frac{-18±\sqrt{144}}{2}$$

Take the square root of 144.

$$x=\frac{-18±12}{2}$$

Now solve the equation x=\frac{-18±12}{2} when ± is plus. Add -18 to 12.

$$x=-\frac{6}{2}$$

Divide -6 by 2.

$$x=-3$$

Now solve the equation x=\frac{-18±12}{2} when ± is minus. Subtract 12 from -18.

$$x=-\frac{30}{2}$$

Divide -30 by 2.

$$x=-15$$

The equation is now solved.

$$x=-3 x=-15$$

Steps for Completing the Square

Take the square root of both sides of the equation.

$$\sqrt{\left(x+9\right)^{2}}=\sqrt{36}$$

Simplify.

$$x+9=6 x+9=-6$$

Subtract 9 from both sides of the equation.

$$x=-3 x=-15$$