Solve sqrt{x+3}+2=sqrt{x+11}

$$\sqrt { x + 3 } + 2 = \sqrt { x + 11 }$$

Solve for x

$$x=-2$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{x+3}+2\right)^{2}=\left(\sqrt{x+11}\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(\sqrt{x+3}+2\right)^{2}.

$$\left(\sqrt{x+3}\right)^{2}+4\sqrt{x+3}+4=\left(\sqrt{x+11}\right)^{2}$$

Calculate \sqrt{x+3} to the power of 2 and get x+3.

$$x+3+4\sqrt{x+3}+4=\left(\sqrt{x+11}\right)^{2}$$

Add 3 and 4 to get 7.

$$x+7+4\sqrt{x+3}=\left(\sqrt{x+11}\right)^{2}$$

Calculate \sqrt{x+11} to the power of 2 and get x+11.

$$x+7+4\sqrt{x+3}=x+11$$

Subtract x from both sides.

$$x+7+4\sqrt{x+3}-x=11$$

Combine x and -x to get 0.

$$7+4\sqrt{x+3}=11$$

Subtract 7 from both sides.

$$4\sqrt{x+3}=11-7$$

Subtract 7 from 11 to get 4.

$$4\sqrt{x+3}=4$$

Divide both sides by 4.

$$\sqrt{x+3}=\frac{4}{4}$$

Divide 4 by 4 to get 1.

$$\sqrt{x+3}=1$$

Square both sides of the equation.

$$x+3=1$$

Subtract 3 from both sides of the equation.

$$x+3-3=1-3$$

Subtracting 3 from itself leaves 0.

$$x=1-3$$

Subtract 3 from 1.

$$x=-2$$

Substitute -2 for x in the equation \sqrt{x+3}+2=\sqrt{x+11}.

$$\sqrt{-2+3}+2=\sqrt{-2+11}$$

Simplify. The value x=-2 satisfies the equation.

$$3=3$$

Equation \sqrt{x+3}+2=\sqrt{x+11} has a unique solution.

$$x=-2$$