Solve sqrt{x-9}+2=sqrt{x-1}

$$\sqrt{ x-9 } +2= \sqrt{ x-1 }$$

Solve for x

$$x=10$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{x-9}+2\right)^{2}=\left(\sqrt{x-1}\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(\sqrt{x-9}+2\right)^{2}.

$$\left(\sqrt{x-9}\right)^{2}+4\sqrt{x-9}+4=\left(\sqrt{x-1}\right)^{2}$$

Calculate \sqrt{x-9} to the power of 2 and get x-9.

$$x-9+4\sqrt{x-9}+4=\left(\sqrt{x-1}\right)^{2}$$

Add -9 and 4 to get -5.

$$x-5+4\sqrt{x-9}=\left(\sqrt{x-1}\right)^{2}$$

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

$$x-5+4\sqrt{x-9}=x-1$$

Subtract x from both sides.

$$x-5+4\sqrt{x-9}-x=-1$$

Combine x and -x to get 0.

$$-5+4\sqrt{x-9}=-1$$

Add 5 to both sides.

$$4\sqrt{x-9}=-1+5$$

Add -1 and 5 to get 4.

$$4\sqrt{x-9}=4$$

Divide both sides by 4.

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

Divide 4 by 4 to get 1.

$$\sqrt{x-9}=1$$

Square both sides of the equation.

$$x-9=1$$

Add 9 to both sides of the equation.

$$x-9-\left(-9\right)=1-\left(-9\right)$$

Subtracting -9 from itself leaves 0.

$$x=1-\left(-9\right)$$

Subtract -9 from 1.

$$x=10$$

Substitute 10 for x in the equation \sqrt{x-9}+2=\sqrt{x-1}.

$$\sqrt{10-9}+2=\sqrt{10-1}$$

Simplify. The value x=10 satisfies the equation.

$$3=3$$

Equation \sqrt{x-9}+2=\sqrt{x-1} has a unique solution.

$$x=10$$