Solve sqrt{x+3}-sqrt{x+8}+1=0

$$\sqrt { x + 3 } - \sqrt { x + 8 } + 1 = 0$$

Solve for x

$$x=1$$

Solution Steps

Subtract -\sqrt{x+8}+1 from both sides of the equation.

$$\sqrt{x+3}=-\left(-\sqrt{x+8}+1\right)$$

To find the opposite of -\sqrt{x+8}+1, find the opposite of each term.

$$\sqrt{x+3}=-\left(-\sqrt{x+8}\right)-1$$

The opposite of -\sqrt{x+8} is \sqrt{x+8}.

$$\sqrt{x+3}=\sqrt{x+8}-1$$

Square both sides of the equation.

$$\left(\sqrt{x+3}\right)^{2}=\left(\sqrt{x+8}-1\right)^{2}$$

Calculate \sqrt{x+3} to the power of 2 and get x+3.

$$x+3=\left(\sqrt{x+8}-1\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\sqrt{x+8}-1\right)^{2}.

$$x+3=\left(\sqrt{x+8}\right)^{2}-2\sqrt{x+8}+1$$

Calculate \sqrt{x+8} to the power of 2 and get x+8.

$$x+3=x+8-2\sqrt{x+8}+1$$

Add 8 and 1 to get 9.

$$x+3=x+9-2\sqrt{x+8}$$

Subtract x from both sides.

$$x+3-x=9-2\sqrt{x+8}$$

Combine x and -x to get 0.

$$3=9-2\sqrt{x+8}$$

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

$$9-2\sqrt{x+8}=3$$

Subtract 9 from both sides.

$$-2\sqrt{x+8}=3-9$$

Subtract 9 from 3 to get -6.

$$-2\sqrt{x+8}=-6$$

Divide both sides by -2.

$$\sqrt{x+8}=\frac{-6}{-2}$$

Divide -6 by -2 to get 3.

$$\sqrt{x+8}=3$$

Square both sides of the equation.

$$x+8=9$$

Subtract 8 from both sides of the equation.

$$x+8-8=9-8$$

Subtracting 8 from itself leaves 0.

$$x=9-8$$

Subtract 8 from 9.

$$x=1$$

Substitute 1 for x in the equation \sqrt{x+3}-\sqrt{x+8}+1=0.

$$\sqrt{1+3}-\sqrt{1+8}+1=0$$

Simplify. The value x=1 satisfies the equation.

$$0=0$$

Equation \sqrt{x+3}=\sqrt{x+8}-1 has a unique solution.

$$x=1$$