Solve sqrt{2x+4}-sqrt{x+9}=1

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

Solve for x

$$x=16$$

Solution Steps

Subtract -\sqrt{x+9} from both sides of the equation.

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

Square both sides of the equation.

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

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

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

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

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

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

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

Add 1 and 9 to get 10.

$$2x+4=10+2\sqrt{x+9}+x$$

Subtract 10+x from both sides of the equation.

$$2x+4-\left(10+x\right)=2\sqrt{x+9}$$

To find the opposite of 10+x, find the opposite of each term.

$$2x+4-10-x=2\sqrt{x+9}$$

Subtract 10 from 4 to get -6.

$$2x-6-x=2\sqrt{x+9}$$

Combine 2x and -x to get x.

$$x-6=2\sqrt{x+9}$$

Square both sides of the equation.

$$\left(x-6\right)^{2}=\left(2\sqrt{x+9}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-6\right)^{2}.

$$x^{2}-12x+36=\left(2\sqrt{x+9}\right)^{2}$$

Expand \left(2\sqrt{x+9}\right)^{2}.

$$x^{2}-12x+36=2^{2}\left(\sqrt{x+9}\right)^{2}$$

Calculate 2 to the power of 2 and get 4.

$$x^{2}-12x+36=4\left(\sqrt{x+9}\right)^{2}$$

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

$$x^{2}-12x+36=4\left(x+9\right)$$

Use the distributive property to multiply 4 by x+9.

$$x^{2}-12x+36=4x+36$$

Subtract 4x from both sides.

$$x^{2}-12x+36-4x=36$$

Combine -12x and -4x to get -16x.

$$x^{2}-16x+36=36$$

Subtract 36 from both sides.

$$x^{2}-16x+36-36=0$$

Subtract 36 from 36 to get 0.

$$x^{2}-16x=0$$

Factor out x.

$$x\left(x-16\right)=0$$

To find equation solutions, solve x=0 and x-16=0.

$$x=0 x=16$$

Substitute 0 for x in the equation \sqrt{2x+4}-\sqrt{x+9}=1.

$$\sqrt{2\times 0+4}-\sqrt{0+9}=1$$

Simplify. The value x=0 does not satisfy the equation because the left and the right hand side have opposite signs.

$$-1=1$$

Substitute 16 for x in the equation \sqrt{2x+4}-\sqrt{x+9}=1.

$$\sqrt{2\times 16+4}-\sqrt{16+9}=1$$

Simplify. The value x=16 satisfies the equation.

$$1=1$$

Equation \sqrt{2x+4}=\sqrt{x+9}+1 has a unique solution.

$$x=16$$