Solve sqrt{n+18}=n-2

$$\sqrt { n + 18 } = n - 2$$

Solve for n

$$n=7$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{n+18}\right)^{2}=\left(n-2\right)^{2}$$

Calculate \sqrt{n+18} to the power of 2 and get n+18.

$$n+18=\left(n-2\right)^{2}$$

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

$$n+18=n^{2}-4n+4$$

Subtract n^{2} from both sides.

$$n+18-n^{2}=-4n+4$$

Add 4n to both sides.

$$n+18-n^{2}+4n=4$$

Combine n and 4n to get 5n.

$$5n+18-n^{2}=4$$

Subtract 4 from both sides.

$$5n+18-n^{2}-4=0$$

Subtract 4 from 18 to get 14.

$$5n+14-n^{2}=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$-n^{2}+5n+14=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -n^{2}+an+bn+14. To find a and b, set up a system to be solved.

$$a+b=5 ab=-14=-14$$

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 -14.

$$-1,14 -2,7$$

Calculate the sum for each pair.

$$-1+14=13 -2+7=5$$

The solution is the pair that gives sum 5.

$$a=7 b=-2$$

Rewrite -n^{2}+5n+14 as \left(-n^{2}+7n\right)+\left(-2n+14\right).

$$\left(-n^{2}+7n\right)+\left(-2n+14\right)$$

Factor out -n in the first and -2 in the second group.

$$-n\left(n-7\right)-2\left(n-7\right)$$

Factor out common term n-7 by using distributive property.

$$\left(n-7\right)\left(-n-2\right)$$

To find equation solutions, solve n-7=0 and -n-2=0.

$$n=7 n=-2$$

Substitute 7 for n in the equation \sqrt{n+18}=n-2.

$$\sqrt{7+18}=7-2$$

Simplify. The value n=7 satisfies the equation.

$$5=5$$

Substitute -2 for n in the equation \sqrt{n+18}=n-2.

$$\sqrt{-2+18}=-2-2$$

Simplify. The value n=-2 does not satisfy the equation because the left and the right hand side have opposite signs.

$$4=-4$$

Equation \sqrt{n+18}=n-2 has a unique solution.

$$n=7$$