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

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

Solve for x

$$x = \frac{5}{4} = 1\frac{1}{4} = 1.25$$

Solution Steps

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

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

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

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

The opposite of -\sqrt{x+1} is \sqrt{x+1}.

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

Square both sides of the equation.

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

Calculate \sqrt{x-1} to the power of 2 and get x-1.

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

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

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

Calculate \sqrt{x+1} to the power of 2 and get x+1.

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

Add 1 and 1 to get 2.

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

Subtract x from both sides.

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

Combine x and -x to get 0.

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

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

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

Subtract 2 from both sides.

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

Subtract 2 from -1 to get -3.

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

Divide both sides by -2.

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

Fraction \frac{-3}{-2} can be simplified to \frac{3}{2} by removing the negative sign from both the numerator and the denominator.

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

Square both sides of the equation.

$$x+1=\frac{9}{4}$$

Subtract 1 from both sides of the equation.

$$x+1-1=\frac{9}{4}-1$$

Subtracting 1 from itself leaves 0.

$$x=\frac{9}{4}-1$$

Subtract 1 from \frac{9}{4}.

$$x=\frac{5}{4}$$

Substitute \frac{5}{4} for x in the equation \sqrt{x-1}-\sqrt{x+1}+1=0.

$$\sqrt{\frac{5}{4}-1}-\sqrt{\frac{5}{4}+1}+1=0$$

Simplify. The value x=\frac{5}{4} satisfies the equation.

$$0=0$$

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

$$x=\frac{5}{4}$$