Solve sqrt{x-2}-sqrt{x+3}+1=0

$$\sqrt{ x-2 } - \sqrt{ x+3 } +1 = 0$$

Solve for x

$$x=6$$

Solution Steps

Subtract -\sqrt{x+3}+1 from both sides of the equation.

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

To find the opposite of -\sqrt{x+3}+1, find the opposite of each term.

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

The opposite of -\sqrt{x+3} is \sqrt{x+3}.

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

Square both sides of the equation.

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

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

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(\sqrt{x+3}-1\right)^{2}.

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

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

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

Add 3 and 1 to get 4.

$$x-2=x+4-2\sqrt{x+3}$$

Subtract x from both sides.

$$x-2-x=4-2\sqrt{x+3}$$

Combine x and -x to get 0.

$$-2=4-2\sqrt{x+3}$$

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

$$4-2\sqrt{x+3}=-2$$

Subtract 4 from both sides.

$$-2\sqrt{x+3}=-2-4$$

Subtract 4 from -2 to get -6.

$$-2\sqrt{x+3}=-6$$

Divide both sides by -2.

$$\sqrt{x+3}=\frac{-6}{-2}$$

Divide -6 by -2 to get 3.

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

Square both sides of the equation.

$$x+3=9$$

Subtract 3 from both sides of the equation.

$$x+3-3=9-3$$

Subtracting 3 from itself leaves 0.

$$x=9-3$$

Subtract 3 from 9.

$$x=6$$

Substitute 6 for x in the equation \sqrt{x-2}-\sqrt{x+3}+1=0.

$$\sqrt{6-2}-\sqrt{6+3}+1=0$$

Simplify. The value x=6 satisfies the equation.

$$0=0$$

Equation \sqrt{x-2}=\sqrt{x+3}-1 has a unique solution.

$$x=6$$