Solve sqrt{x+8}=2+x

$$\sqrt { x + 8 } = 2 + x$$

Solve for x

$$x=1$$

Solution Steps

Square both sides of the equation.

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

Calculate \sqrt{x+8} to the power of 2 and get x+8.

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

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

$$x+8=4+4x+x^{2}$$

Subtract 4 from both sides.

$$x+8-4=4x+x^{2}$$

Subtract 4 from 8 to get 4.

$$x+4=4x+x^{2}$$

Subtract 4x from both sides.

$$x+4-4x=x^{2}$$

Combine x and -4x to get -3x.

$$-3x+4=x^{2}$$

Subtract x^{2} from both sides.

$$-3x+4-x^{2}=0$$

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

$$-x^{2}-3x+4=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx+4. To find a and b, set up a system to be solved.

$$a+b=-3 ab=-4=-4$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -4.

$$1,-4 2,-2$$

Calculate the sum for each pair.

$$1-4=-3 2-2=0$$

The solution is the pair that gives sum -3.

$$a=1 b=-4$$

Rewrite -x^{2}-3x+4 as \left(-x^{2}+x\right)+\left(-4x+4\right).

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

Factor out x in the first and 4 in the second group.

$$x\left(-x+1\right)+4\left(-x+1\right)$$

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

$$\left(-x+1\right)\left(x+4\right)$$

To find equation solutions, solve -x+1=0 and x+4=0.

$$x=1 x=-4$$

Substitute 1 for x in the equation \sqrt{x+8}=2+x.

$$\sqrt{1+8}=2+1$$

Simplify. The value x=1 satisfies the equation.

$$3=3$$

Substitute -4 for x in the equation \sqrt{x+8}=2+x.

$$\sqrt{-4+8}=2-4$$

Simplify. The value x=-4 does not satisfy the equation because the left and the right hand side have opposite signs.

$$2=-2$$

Equation \sqrt{x+8}=x+2 has a unique solution.

$$x=1$$