Solve 9(x+8)^2=81

$$9 ( x + 8 ) ^ { 2 } = 81$$

Solve for x

$$x=-5$$

$$x=-11$$

Steps Using Factoring

Divide both sides by 9.

$$\left(x+8\right)^{2}=\frac{81}{9}$$

Divide 81 by 9 to get 9.

$$\left(x+8\right)^{2}=9$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+8\right)^{2}.

$$x^{2}+16x+64=9$$

Subtract 9 from both sides.

$$x^{2}+16x+64-9=0$$

Subtract 9 from 64 to get 55.

$$x^{2}+16x+55=0$$

To solve the equation, factor x^{2}+16x+55 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.

$$a+b=16 ab=55$$

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 55.

$$1,55 5,11$$

Calculate the sum for each pair.

$$1+55=56 5+11=16$$

The solution is the pair that gives sum 16.

$$a=5 b=11$$

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

$$\left(x+5\right)\left(x+11\right)$$

To find equation solutions, solve x+5=0 and x+11=0.

$$x=-5 x=-11$$

Steps Using Factoring By Grouping

Divide both sides by 9.

$$\left(x+8\right)^{2}=\frac{81}{9}$$

Divide 81 by 9 to get 9.

$$\left(x+8\right)^{2}=9$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+8\right)^{2}.

$$x^{2}+16x+64=9$$

Subtract 9 from both sides.

$$x^{2}+16x+64-9=0$$

Subtract 9 from 64 to get 55.

$$x^{2}+16x+55=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx+55. To find a and b, set up a system to be solved.

$$a+b=16 ab=1\times 55=55$$

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 55.

$$1,55 5,11$$

Calculate the sum for each pair.

$$1+55=56 5+11=16$$

The solution is the pair that gives sum 16.

$$a=5 b=11$$

Rewrite x^{2}+16x+55 as \left(x^{2}+5x\right)+\left(11x+55\right).

$$\left(x^{2}+5x\right)+\left(11x+55\right)$$

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

$$x\left(x+5\right)+11\left(x+5\right)$$

Factor out common term x+5 by using distributive property.

$$\left(x+5\right)\left(x+11\right)$$

To find equation solutions, solve x+5=0 and x+11=0.

$$x=-5 x=-11$$

Steps Using the Quadratic Formula

Divide both sides by 9.

$$\left(x+8\right)^{2}=\frac{81}{9}$$

Divide 81 by 9 to get 9.

$$\left(x+8\right)^{2}=9$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+8\right)^{2}.

$$x^{2}+16x+64=9$$

Subtract 9 from both sides.

$$x^{2}+16x+64-9=0$$

Subtract 9 from 64 to get 55.

$$x^{2}+16x+55=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 16 for b, and 55 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-16±\sqrt{16^{2}-4\times 55}}{2}$$

Square 16.

$$x=\frac{-16±\sqrt{256-4\times 55}}{2}$$

Multiply -4 times 55.

$$x=\frac{-16±\sqrt{256-220}}{2}$$

Add 256 to -220.

$$x=\frac{-16±\sqrt{36}}{2}$$

Take the square root of 36.

$$x=\frac{-16±6}{2}$$

Now solve the equation x=\frac{-16±6}{2} when ± is plus. Add -16 to 6.

$$x=-\frac{10}{2}$$

Divide -10 by 2.

$$x=-5$$

Now solve the equation x=\frac{-16±6}{2} when ± is minus. Subtract 6 from -16.

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

Divide -22 by 2.

$$x=-11$$

The equation is now solved.

$$x=-5 x=-11$$

Steps for Completing the Square

Divide both sides by 9.

$$\left(x+8\right)^{2}=\frac{81}{9}$$

Divide 81 by 9 to get 9.

$$\left(x+8\right)^{2}=9$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+8\right)^{2}}=\sqrt{9}$$

Simplify.

$$x+8=3 x+8=-3$$

Subtract 8 from both sides of the equation.

$$x=-5 x=-11$$