Solve sqrt{x+4}=1+sqrt{x-9}

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

Solve for x

$$x=45$$

Solution Steps

Square both sides of the equation.

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

Calculate \sqrt{x+4} to the power of 2 and get x+4.

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

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(1+\sqrt{x-9}\right)^{2}.

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

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

$$x+4=1+2\sqrt{x-9}+x-9$$

Subtract 9 from 1 to get -8.

$$x+4=-8+2\sqrt{x-9}+x$$

Subtract 2\sqrt{x-9} from both sides.

$$x+4-2\sqrt{x-9}=-8+x$$

Subtract x from both sides.

$$x+4-2\sqrt{x-9}-x=-8$$

Combine x and -x to get 0.

$$4-2\sqrt{x-9}=-8$$

Subtract 4 from both sides.

$$-2\sqrt{x-9}=-8-4$$

Subtract 4 from -8 to get -12.

$$-2\sqrt{x-9}=-12$$

Divide both sides by -2.

$$\sqrt{x-9}=\frac{-12}{-2}$$

Divide -12 by -2 to get 6.

$$\sqrt{x-9}=6$$

Square both sides of the equation.

$$x-9=36$$

Add 9 to both sides of the equation.

$$x-9-\left(-9\right)=36-\left(-9\right)$$

Subtracting -9 from itself leaves 0.

$$x=36-\left(-9\right)$$

Subtract -9 from 36.

$$x=45$$

Substitute 45 for x in the equation \sqrt{x+4}=1+\sqrt{x-9}.

$$\sqrt{45+4}=1+\sqrt{45-9}$$

Simplify. The value x=45 satisfies the equation.

$$7=7$$

Equation \sqrt{x+4}=\sqrt{x-9}+1 has a unique solution.

$$x=45$$