Solve sqrt{x+7}=sqrt{x}+1

$$\sqrt{ x+7 } = \sqrt{ x } +1$$

Solve for x

$$x=9$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{x+7}\right)^{2}=\left(\sqrt{x}+1\right)^{2}$$

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

$$x+7=\left(\sqrt{x}+1\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(\sqrt{x}+1\right)^{2}.

$$x+7=\left(\sqrt{x}\right)^{2}+2\sqrt{x}+1$$

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

$$x+7=x+2\sqrt{x}+1$$

Subtract x from both sides.

$$x+7-x=2\sqrt{x}+1$$

Combine x and -x to get 0.

$$7=2\sqrt{x}+1$$

Swap sides so that all variable terms are on the left hand side.

$$2\sqrt{x}+1=7$$

Subtract 1 from both sides.

$$2\sqrt{x}=7-1$$

Subtract 1 from 7 to get 6.

$$2\sqrt{x}=6$$

Divide both sides by 2.

$$\sqrt{x}=\frac{6}{2}$$

Divide 6 by 2 to get 3.

$$\sqrt{x}=3$$

Square both sides of the equation.

$$x=9$$

Substitute 9 for x in the equation \sqrt{x+7}=\sqrt{x}+1.

$$\sqrt{9+7}=\sqrt{9}+1$$

Simplify. The value x=9 satisfies the equation.

$$4=4$$

Equation \sqrt{x+7}=\sqrt{x}+1 has a unique solution.

$$x=9$$