Solve =sqrt{x^2+24}=x+2

$$= \sqrt{ { x }^{ 2 } +24 } =x+2$$

Solve for x

$$x=5$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{x^{2}+24}\right)^{2}=\left(x+2\right)^{2}$$

Calculate \sqrt{x^{2}+24} to the power of 2 and get x^{2}+24.

$$x^{2}+24=\left(x+2\right)^{2}$$

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

$$x^{2}+24=x^{2}+4x+4$$

Subtract x^{2} from both sides.

$$x^{2}+24-x^{2}=4x+4$$

Combine x^{2} and -x^{2} to get 0.

$$24=4x+4$$

Swap sides so that all variable terms are on the left hand side.

$$4x+4=24$$

Subtract 4 from both sides.

$$4x=24-4$$

Subtract 4 from 24 to get 20.

$$4x=20$$

Divide both sides by 4.

$$x=\frac{20}{4}$$

Divide 20 by 4 to get 5.

$$x=5$$

Substitute 5 for x in the equation \sqrt{x^{2}+24}=x+2.

$$\sqrt{5^{2}+24}=5+2$$

Simplify. The value x=5 satisfies the equation.

$$7=7$$

Equation \sqrt{x^{2}+24}=x+2 has a unique solution.

$$x=5$$