Solve sqrt{100-x}-x=10

$$\sqrt { 100 - x } - x = 10$$

Solve for x

$$x=0$$

Solution Steps

Subtract -x from both sides of the equation.

$$\sqrt{100-x}=10+x$$

Square both sides of the equation.

$$\left(\sqrt{100-x}\right)^{2}=\left(10+x\right)^{2}$$

Calculate \sqrt{100-x} to the power of 2 and get 100-x.

$$100-x=\left(10+x\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(10+x\right)^{2}.

$$100-x=100+20x+x^{2}$$

Subtract 100 from both sides.

$$100-x-100=20x+x^{2}$$

Subtract 100 from 100 to get 0.

$$-x=20x+x^{2}$$

Subtract 20x from both sides.

$$-x-20x=x^{2}$$

Combine -x and -20x to get -21x.

$$-21x=x^{2}$$

Subtract x^{2} from both sides.

$$-21x-x^{2}=0$$

Factor out x.

$$x\left(-21-x\right)=0$$

To find equation solutions, solve x=0 and -21-x=0.

$$x=0 x=-21$$

Substitute 0 for x in the equation \sqrt{100-x}-x=10.

$$\sqrt{100-0}-0=10$$

Simplify. The value x=0 satisfies the equation.

$$10=10$$

Substitute -21 for x in the equation \sqrt{100-x}-x=10.

$$\sqrt{100-\left(-21\right)}-\left(-21\right)=10$$

Simplify. The value x=-21 does not satisfy the equation.

$$32=10$$

Equation \sqrt{100-x}=x+10 has a unique solution.

$$x=0$$