Solve sqrt{x}-1=sqrt{x-10}

$$\sqrt { x } - 1 = \sqrt { x - 10 }$$

Solve for x

$$x = \frac{121}{4} = 30\frac{1}{4} = 30.25$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{x}-1\right)^{2}=\left(\sqrt{x-10}\right)^{2}$$

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

$$\left(\sqrt{x}\right)^{2}-2\sqrt{x}+1=\left(\sqrt{x-10}\right)^{2}$$

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

$$x-2\sqrt{x}+1=\left(\sqrt{x-10}\right)^{2}$$

Calculate \sqrt{x-10} to the power of 2 and get x-10.

$$x-2\sqrt{x}+1=x-10$$

Subtract x from both sides.

$$x-2\sqrt{x}+1-x=-10$$

Combine x and -x to get 0.

$$-2\sqrt{x}+1=-10$$

Subtract 1 from both sides.

$$-2\sqrt{x}=-10-1$$

Subtract 1 from -10 to get -11.

$$-2\sqrt{x}=-11$$

Divide both sides by -2.

$$\sqrt{x}=\frac{-11}{-2}$$

Fraction \frac{-11}{-2} can be simplified to \frac{11}{2} by removing the negative sign from both the numerator and the denominator.

$$\sqrt{x}=\frac{11}{2}$$

Square both sides of the equation.

$$x=\frac{121}{4}$$

Substitute \frac{121}{4} for x in the equation \sqrt{x}-1=\sqrt{x-10}.

$$\sqrt{\frac{121}{4}}-1=\sqrt{\frac{121}{4}-10}$$

Simplify. The value x=\frac{121}{4} satisfies the equation.

$$\frac{9}{2}=\frac{9}{2}$$

Equation \sqrt{x}-1=\sqrt{x-10} has a unique solution.

$$x=\frac{121}{4}$$