Solve sqrt{x}-2=sqrt{x-56}

$$\sqrt { x } - 2 = \sqrt { x - 56 }$$

Solve for x

$$x=225$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{x}-2\right)^{2}=\left(\sqrt{x-56}\right)^{2}$$

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

$$\left(\sqrt{x}\right)^{2}-4\sqrt{x}+4=\left(\sqrt{x-56}\right)^{2}$$

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

$$x-4\sqrt{x}+4=\left(\sqrt{x-56}\right)^{2}$$

Calculate \sqrt{x-56} to the power of 2 and get x-56.

$$x-4\sqrt{x}+4=x-56$$

Subtract x from both sides.

$$x-4\sqrt{x}+4-x=-56$$

Combine x and -x to get 0.

$$-4\sqrt{x}+4=-56$$

Subtract 4 from both sides.

$$-4\sqrt{x}=-56-4$$

Subtract 4 from -56 to get -60.

$$-4\sqrt{x}=-60$$

Divide both sides by -4.

$$\sqrt{x}=\frac{-60}{-4}$$

Divide -60 by -4 to get 15.

$$\sqrt{x}=15$$

Square both sides of the equation.

$$x=225$$

Substitute 225 for x in the equation \sqrt{x}-2=\sqrt{x-56}.

$$\sqrt{225}-2=\sqrt{225-56}$$

Simplify. The value x=225 satisfies the equation.

$$13=13$$

Equation \sqrt{x}-2=\sqrt{x-56} has a unique solution.

$$x=225$$