Solve sqrt{x^2-13}+x=13

$$\sqrt { x ^ { 2 } - 13 } + x = 13$$

Solve for x

$$x=7$$

Solution Steps

Subtract x from both sides of the equation.

$$\sqrt{x^{2}-13}=13-x$$

Square both sides of the equation.

$$\left(\sqrt{x^{2}-13}\right)^{2}=\left(13-x\right)^{2}$$

Calculate \sqrt{x^{2}-13} to the power of 2 and get x^{2}-13.

$$x^{2}-13=\left(13-x\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(13-x\right)^{2}.

$$x^{2}-13=169-26x+x^{2}$$

Add 26x to both sides.

$$x^{2}-13+26x=169+x^{2}$$

Subtract x^{2} from both sides.

$$x^{2}-13+26x-x^{2}=169$$

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

$$-13+26x=169$$

Add 13 to both sides.

$$26x=169+13$$

Add 169 and 13 to get 182.

$$26x=182$$

Divide both sides by 26.

$$x=\frac{182}{26}$$

Divide 182 by 26 to get 7.

$$x=7$$

Substitute 7 for x in the equation \sqrt{x^{2}-13}+x=13.

$$\sqrt{7^{2}-13}+7=13$$

Simplify. The value x=7 satisfies the equation.

$$13=13$$

Equation \sqrt{x^{2}-13}=13-x has a unique solution.

$$x=7$$