Solve n^2+16n+64=36

$$n ^ { 2 } + 16 n + 64 = 36$$

Solve for n

$$n=-14$$

$$n=-2$$

Steps Using Factoring

Subtract 36 from both sides.

$$n^{2}+16n+64-36=0$$

Subtract 36 from 64 to get 28.

$$n^{2}+16n+28=0$$

To solve the equation, factor n^{2}+16n+28 using formula n^{2}+\left(a+b\right)n+ab=\left(n+a\right)\left(n+b\right). To find a and b, set up a system to be solved.

$$a+b=16 ab=28$$

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 28.

$$1,28 2,14 4,7$$

Calculate the sum for each pair.

$$1+28=29 2+14=16 4+7=11$$

The solution is the pair that gives sum 16.

$$a=2 b=14$$

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

$$\left(n+2\right)\left(n+14\right)$$

To find equation solutions, solve n+2=0 and n+14=0.

$$n=-2 n=-14$$

Steps Using Factoring By Grouping

Subtract 36 from both sides.

$$n^{2}+16n+64-36=0$$

Subtract 36 from 64 to get 28.

$$n^{2}+16n+28=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as n^{2}+an+bn+28. To find a and b, set up a system to be solved.

$$a+b=16 ab=1\times 28=28$$

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 28.

$$1,28 2,14 4,7$$

Calculate the sum for each pair.

$$1+28=29 2+14=16 4+7=11$$

The solution is the pair that gives sum 16.

$$a=2 b=14$$

Rewrite n^{2}+16n+28 as \left(n^{2}+2n\right)+\left(14n+28\right).

$$\left(n^{2}+2n\right)+\left(14n+28\right)$$

Factor out n in the first and 14 in the second group.

$$n\left(n+2\right)+14\left(n+2\right)$$

Factor out common term n+2 by using distributive property.

$$\left(n+2\right)\left(n+14\right)$$

To find equation solutions, solve n+2=0 and n+14=0.

$$n=-2 n=-14$$

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.

$$n^{2}+16n+64=36$$

Subtract 36 from both sides of the equation.

$$n^{2}+16n+64-36=36-36$$

Subtracting 36 from itself leaves 0.

$$n^{2}+16n+64-36=0$$

Subtract 36 from 64.

$$n^{2}+16n+28=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 16 for b, and 28 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-16±\sqrt{16^{2}-4\times 28}}{2}$$

Square 16.

$$n=\frac{-16±\sqrt{256-4\times 28}}{2}$$

Multiply -4 times 28.

$$n=\frac{-16±\sqrt{256-112}}{2}$$

Add 256 to -112.

$$n=\frac{-16±\sqrt{144}}{2}$$

Take the square root of 144.

$$n=\frac{-16±12}{2}$$

Now solve the equation n=\frac{-16±12}{2} when ± is plus. Add -16 to 12.

$$n=-\frac{4}{2}$$

Divide -4 by 2.

$$n=-2$$

Now solve the equation n=\frac{-16±12}{2} when ± is minus. Subtract 12 from -16.

$$n=-\frac{28}{2}$$

Divide -28 by 2.

$$n=-14$$

The equation is now solved.

$$n=-2 n=-14$$

Steps for Completing the Square

Factor n^{2}+16n+64. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(n+8\right)^{2}=36$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n+8\right)^{2}}=\sqrt{36}$$

Simplify.

$$n+8=6 n+8=-6$$

Subtract 8 from both sides of the equation.

$$n=-2 n=-14$$