Solve x+sqrt{(x+2)}=18

$$x+ \sqrt{ (x+2) } =18$$

Solve for x

$$x=14$$

Solution Steps

Subtract x from both sides of the equation.

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

Square both sides of the equation.

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

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

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

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

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

Subtract 324 from both sides.

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

Subtract 324 from 2 to get -322.

$$x-322=-36x+x^{2}$$

Add 36x to both sides.

$$x-322+36x=x^{2}$$

Combine x and 36x to get 37x.

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

Subtract x^{2} from both sides.

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

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 x+\sqrt{x+2}=18.

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

Simplify. The value x=23 does not satisfy the equation.

$$28=18$$

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

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

Simplify. The value x=14 satisfies the equation.

$$18=18$$

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

$$x=14$$