Solve sqrt{x+2}=x-18

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

Solve for x

$$x=23$$

Solution Steps

Square both sides of the equation.

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

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

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-18\right)^{2}.

$$x+2=x^{2}-36x+324$$

Subtract x^{2} from both sides.

$$x+2-x^{2}=-36x+324$$

Add 36x to both sides.

$$x+2-x^{2}+36x=324$$

Combine x and 36x to get 37x.

$$37x+2-x^{2}=324$$

Subtract 324 from both sides.

$$37x+2-x^{2}-324=0$$

Subtract 324 from 2 to get -322.

$$37x-322-x^{2}=0$$

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

$$-x^{2}+37x-322=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-322. To find a and b, set up a system to be solved.

$$a+b=37 ab=-\left(-322\right)=322$$

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 322.

$$1,322 2,161 7,46 14,23$$

Calculate the sum for each pair.

$$1+322=323 2+161=163 7+46=53 14+23=37$$

The solution is the pair that gives sum 37.

$$a=23 b=14$$

Rewrite -x^{2}+37x-322 as \left(-x^{2}+23x\right)+\left(14x-322\right).

$$\left(-x^{2}+23x\right)+\left(14x-322\right)$$

Factor out -x in the first and 14 in the second group.

$$-x\left(x-23\right)+14\left(x-23\right)$$

Factor out common term x-23 by using distributive property.

$$\left(x-23\right)\left(-x+14\right)$$

To find equation solutions, solve x-23=0 and -x+14=0.

$$x=23 x=14$$

Substitute 23 for x in the equation \sqrt{x+2}=x-18.

$$\sqrt{23+2}=23-18$$

Simplify. The value x=23 satisfies the equation.

$$5=5$$

Substitute 14 for x in the equation \sqrt{x+2}=x-18.

$$\sqrt{14+2}=14-18$$

Simplify. The value x=14 does not satisfy the equation because the left and the right hand side have opposite signs.

$$4=-4$$

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

$$x=23$$