Solve x^2+8x+16=9

$$x ^ { 2 } + 8 x + 16 = 9$$

Solve for x

$$x=-7$$

$$x=-1$$

Steps Using Factoring

Subtract 9 from both sides.

$$x^{2}+8x+16-9=0$$

Subtract 9 from 16 to get 7.

$$x^{2}+8x+7=0$$

To solve the equation, factor x^{2}+8x+7 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.

$$a+b=8 ab=7$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. The only such pair is the system solution.

$$a=1 b=7$$

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

$$\left(x+1\right)\left(x+7\right)$$

To find equation solutions, solve x+1=0 and x+7=0.

$$x=-1 x=-7$$

Steps Using Factoring By Grouping

Subtract 9 from both sides.

$$x^{2}+8x+16-9=0$$

Subtract 9 from 16 to get 7.

$$x^{2}+8x+7=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx+7. To find a and b, set up a system to be solved.

$$a+b=8 ab=1\times 7=7$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. The only such pair is the system solution.

$$a=1 b=7$$

Rewrite x^{2}+8x+7 as \left(x^{2}+x\right)+\left(7x+7\right).

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

Factor out x in the first and 7 in the second group.

$$x\left(x+1\right)+7\left(x+1\right)$$

Factor out common term x+1 by using distributive property.

$$\left(x+1\right)\left(x+7\right)$$

To find equation solutions, solve x+1=0 and x+7=0.

$$x=-1 x=-7$$

Steps Using the Quadratic Formula

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$x^{2}+8x+16=9$$

Subtract 9 from both sides of the equation.

$$x^{2}+8x+16-9=9-9$$

Subtracting 9 from itself leaves 0.

$$x^{2}+8x+16-9=0$$

Subtract 9 from 16.

$$x^{2}+8x+7=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 8 for b, and 7 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-8±\sqrt{8^{2}-4\times 7}}{2}$$

Square 8.

$$x=\frac{-8±\sqrt{64-4\times 7}}{2}$$

Multiply -4 times 7.

$$x=\frac{-8±\sqrt{64-28}}{2}$$

Add 64 to -28.

$$x=\frac{-8±\sqrt{36}}{2}$$

Take the square root of 36.

$$x=\frac{-8±6}{2}$$

Now solve the equation x=\frac{-8±6}{2} when ± is plus. Add -8 to 6.

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

Divide -2 by 2.

$$x=-1$$

Now solve the equation x=\frac{-8±6}{2} when ± is minus. Subtract 6 from -8.

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

Divide -14 by 2.

$$x=-7$$

The equation is now solved.

$$x=-1 x=-7$$

Steps for Completing the Square

Factor x^{2}+8x+16. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

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

Take the square root of both sides of the equation.

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

Simplify.

$$x+4=3 x+4=-3$$

Subtract 4 from both sides of the equation.

$$x=-1 x=-7$$