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

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

Solve for x

$$x=1$$

Solution Steps

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

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

Square both sides of the equation.

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

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

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

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

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

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

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

Subtract 1 from 1 to get 0.

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

Subtract 2\sqrt{x-1} from both sides.

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

Subtract x from both sides.

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

Combine x and -x to get 0.

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

Divide both sides by -2. Zero divided by any non-zero number gives zero.

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

Square both sides of the equation.

$$x-1=0$$

Add 1 to both sides of the equation.

$$x-1-\left(-1\right)=-\left(-1\right)$$

Subtracting -1 from itself leaves 0.

$$x=-\left(-1\right)$$

Subtract -1 from 0.

$$x=1$$

Substitute 1 for x in the equation \sqrt{x}-\sqrt{x-1}=1.

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

Simplify. The value x=1 satisfies the equation.

$$1=1$$

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

$$x=1$$