Solve (x+8)^2=169

$$( x + 8 ) ^ { 2 } = 169$$

Solve for x

$$x=5$$

$$x=-21$$

Steps Using Factoring

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

$$x^{2}+16x+64=169$$

Subtract 169 from both sides.

$$x^{2}+16x+64-169=0$$

Subtract 169 from 64 to get -105.

$$x^{2}+16x-105=0$$

To solve the equation, factor x^{2}+16x-105 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.

$$a+b=16 ab=-105$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -105.

$$-1,105 -3,35 -5,21 -7,15$$

Calculate the sum for each pair.

$$-1+105=104 -3+35=32 -5+21=16 -7+15=8$$

The solution is the pair that gives sum 16.

$$a=-5 b=21$$

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

$$\left(x-5\right)\left(x+21\right)$$

To find equation solutions, solve x-5=0 and x+21=0.

$$x=5 x=-21$$

Steps Using Factoring By Grouping

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

$$x^{2}+16x+64=169$$

Subtract 169 from both sides.

$$x^{2}+16x+64-169=0$$

Subtract 169 from 64 to get -105.

$$x^{2}+16x-105=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx-105. To find a and b, set up a system to be solved.

$$a+b=16 ab=1\left(-105\right)=-105$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -105.

$$-1,105 -3,35 -5,21 -7,15$$

Calculate the sum for each pair.

$$-1+105=104 -3+35=32 -5+21=16 -7+15=8$$

The solution is the pair that gives sum 16.

$$a=-5 b=21$$

Rewrite x^{2}+16x-105 as \left(x^{2}-5x\right)+\left(21x-105\right).

$$\left(x^{2}-5x\right)+\left(21x-105\right)$$

Factor out x in the first and 21 in the second group.

$$x\left(x-5\right)+21\left(x-5\right)$$

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

$$\left(x-5\right)\left(x+21\right)$$

To find equation solutions, solve x-5=0 and x+21=0.

$$x=5 x=-21$$

Steps Using the Quadratic Formula

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

$$x^{2}+16x+64=169$$

Subtract 169 from both sides.

$$x^{2}+16x+64-169=0$$

Subtract 169 from 64 to get -105.

$$x^{2}+16x-105=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 16 for b, and -105 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-16±\sqrt{16^{2}-4\left(-105\right)}}{2}$$

Square 16.

$$x=\frac{-16±\sqrt{256-4\left(-105\right)}}{2}$$

Multiply -4 times -105.

$$x=\frac{-16±\sqrt{256+420}}{2}$$

Add 256 to 420.

$$x=\frac{-16±\sqrt{676}}{2}$$

Take the square root of 676.

$$x=\frac{-16±26}{2}$$

Now solve the equation x=\frac{-16±26}{2} when ± is plus. Add -16 to 26.

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

Divide 10 by 2.

$$x=5$$

Now solve the equation x=\frac{-16±26}{2} when ± is minus. Subtract 26 from -16.

$$x=-\frac{42}{2}$$

Divide -42 by 2.

$$x=-21$$

The equation is now solved.

$$x=5 x=-21$$

Steps for Completing the Square

Take the square root of both sides of the equation.

$$\sqrt{\left(x+8\right)^{2}}=\sqrt{169}$$

Simplify.

$$x+8=13 x+8=-13$$

Subtract 8 from both sides of the equation.

$$x=5 x=-21$$