Solve sqrt{x}+2=sqrt{x+7}

$$\sqrt { x } + 2 = \sqrt { x + 7 }$$

Solve for x

$$x=\frac{9}{16}=0.5625$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{x}+2\right)^{2}=\left(\sqrt{x+7}\right)^{2}$$

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

$$\left(\sqrt{x}\right)^{2}+4\sqrt{x}+4=\left(\sqrt{x+7}\right)^{2}$$

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

$$x+4\sqrt{x}+4=\left(\sqrt{x+7}\right)^{2}$$

Calculate \sqrt{x+7} to the power of 2 and get x+7.

$$x+4\sqrt{x}+4=x+7$$

Subtract x from both sides.

$$x+4\sqrt{x}+4-x=7$$

Combine x and -x to get 0.

$$4\sqrt{x}+4=7$$

Subtract 4 from both sides.

$$4\sqrt{x}=7-4$$

Subtract 4 from 7 to get 3.

$$4\sqrt{x}=3$$

Divide both sides by 4.

$$\sqrt{x}=\frac{3}{4}$$

Square both sides of the equation.

$$x=\frac{9}{16}$$

Substitute \frac{9}{16} for x in the equation \sqrt{x}+2=\sqrt{x+7}.

$$\sqrt{\frac{9}{16}}+2=\sqrt{\frac{9}{16}+7}$$

Simplify. The value x=\frac{9}{16} satisfies the equation.

$$\frac{11}{4}=\frac{11}{4}$$

Equation \sqrt{x}+2=\sqrt{x+7} has a unique solution.

$$x=\frac{9}{16}$$