Solve sqrt{x-95}+sqrt{x}=19

$$\sqrt { x - 95 } + \sqrt { x } = 19$$

Solve for x

$$x=144$$

Solution Steps

Subtract \sqrt{x} from both sides of the equation.

$$\sqrt{x-95}=19-\sqrt{x}$$

Square both sides of the equation.

$$\left(\sqrt{x-95}\right)^{2}=\left(19-\sqrt{x}\right)^{2}$$

Calculate \sqrt{x-95} to the power of 2 and get x-95.

$$x-95=\left(19-\sqrt{x}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(19-\sqrt{x}\right)^{2}.

$$x-95=361-38\sqrt{x}+\left(\sqrt{x}\right)^{2}$$

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

$$x-95=361-38\sqrt{x}+x$$

Add 38\sqrt{x} to both sides.

$$x-95+38\sqrt{x}=361+x$$

Subtract x from both sides.

$$x-95+38\sqrt{x}-x=361$$

Combine x and -x to get 0.

$$-95+38\sqrt{x}=361$$

Add 95 to both sides.

$$38\sqrt{x}=361+95$$

Add 361 and 95 to get 456.

$$38\sqrt{x}=456$$

Divide both sides by 38.

$$\sqrt{x}=\frac{456}{38}$$

Divide 456 by 38 to get 12.

$$\sqrt{x}=12$$

Square both sides of the equation.

$$x=144$$

Substitute 144 for x in the equation \sqrt{x-95}+\sqrt{x}=19.

$$\sqrt{144-95}+\sqrt{144}=19$$

Simplify. The value x=144 satisfies the equation.

$$19=19$$

Equation \sqrt{x-95}=-\sqrt{x}+19 has a unique solution.

$$x=144$$