Solve 5+15=x-5+sqrt{x^2+15^2}

$$5+15=x-5+ \sqrt{ { x }^{ 2 } + { 15 }^{ 2 } }$$

Solve for x

$$x=8$$

Solution Steps

Subtract x-5 from both sides of the equation.

$$5+15-\left(x-5\right)=\sqrt{x^{2}+15^{2}}$$

Add 5 and 15 to get 20.

$$20-\left(x-5\right)=\sqrt{x^{2}+15^{2}}$$

To find the opposite of x-5, find the opposite of each term.

$$20-x+5=\sqrt{x^{2}+15^{2}}$$

Add 20 and 5 to get 25.

$$25-x=\sqrt{x^{2}+15^{2}}$$

Calculate 15 to the power of 2 and get 225.

$$25-x=\sqrt{x^{2}+225}$$

Square both sides of the equation.

$$\left(25-x\right)^{2}=\left(\sqrt{x^{2}+225}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(25-x\right)^{2}.

$$625-50x+x^{2}=\left(\sqrt{x^{2}+225}\right)^{2}$$

Calculate \sqrt{x^{2}+225} to the power of 2 and get x^{2}+225.

$$625-50x+x^{2}=x^{2}+225$$

Subtract x^{2} from both sides.

$$625-50x+x^{2}-x^{2}=225$$

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

$$625-50x=225$$

Subtract 625 from both sides.

$$-50x=225-625$$

Subtract 625 from 225 to get -400.

$$-50x=-400$$

Divide both sides by -50.

$$x=\frac{-400}{-50}$$

Divide -400 by -50 to get 8.

$$x=8$$

Substitute 8 for x in the equation 5+15=x-5+\sqrt{x^{2}+15^{2}}.

$$5+15=8-5+\sqrt{8^{2}+15^{2}}$$

Simplify. The value x=8 satisfies the equation.

$$20=20$$

Equation 25-x=\sqrt{x^{2}+225} has a unique solution.

$$x=8$$