Solve sqrt{x+9}-sqrt{2x}=3

$$\sqrt { x + 9 } - \sqrt { 2 x } = 3$$

Solve for x

$$x=0$$

Solution Steps

Subtract -\sqrt{2x} from both sides of the equation.

$$\sqrt{x+9}=3+\sqrt{2x}$$

Square both sides of the equation.

$$\left(\sqrt{x+9}\right)^{2}=\left(3+\sqrt{2x}\right)^{2}$$

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

$$x+9=\left(3+\sqrt{2x}\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(3+\sqrt{2x}\right)^{2}.

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

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

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

Subtract 9+2x from both sides of the equation.

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

To find the opposite of 9+2x, find the opposite of each term.

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

Subtract 9 from 9 to get 0.

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

Combine x and -2x to get -x.

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

Square both sides of the equation.

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

Expand \left(-x\right)^{2}.

$$\left(-1\right)^{2}x^{2}=\left(6\sqrt{2x}\right)^{2}$$

Calculate -1 to the power of 2 and get 1.

$$1x^{2}=\left(6\sqrt{2x}\right)^{2}$$

Expand \left(6\sqrt{2x}\right)^{2}.

$$1x^{2}=6^{2}\left(\sqrt{2x}\right)^{2}$$

Calculate 6 to the power of 2 and get 36.

$$1x^{2}=36\left(\sqrt{2x}\right)^{2}$$

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

$$1x^{2}=36\times 2x$$

Multiply 36 and 2 to get 72.

$$1x^{2}=72x$$

Reorder the terms.

$$x^{2}=72x$$

Subtract 72x from both sides.

$$x^{2}-72x=0$$

Factor out x.

$$x\left(x-72\right)=0$$

To find equation solutions, solve x=0 and x-72=0.

$$x=0 x=72$$

Substitute 0 for x in the equation \sqrt{x+9}-\sqrt{2x}=3.

$$\sqrt{0+9}-\sqrt{2\times 0}=3$$

Simplify. The value x=0 satisfies the equation.

$$3=3$$

Substitute 72 for x in the equation \sqrt{x+9}-\sqrt{2x}=3.

$$\sqrt{72+9}-\sqrt{2\times 72}=3$$

Simplify. The value x=72 does not satisfy the equation because the left and the right hand side have opposite signs.

$$-3=3$$

Substitute 0 for x in the equation \sqrt{x+9}-\sqrt{2x}=3.

$$\sqrt{0+9}-\sqrt{2\times 0}=3$$

Simplify. The value x=0 satisfies the equation.

$$3=3$$

Equation \sqrt{x+9}=\sqrt{2x}+3 has a unique solution.

$$x=0$$