Solve 2sqrt{x}-x-2=0

$$2 \sqrt { x } - x - 2 = 0$$

Solve for x (complex solution)

$$x=-2i$$

$$x=2i$$

Solution Steps

Subtract -x-2 from both sides of the equation.

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

To find the opposite of -x-2, find the opposite of each term.

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

The opposite of -x is x.

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

The opposite of -2 is 2.

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

Square both sides of the equation.

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

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

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

Calculate 2 to the power of 2 and get 4.

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

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

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

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

$$4x=x^{2}+4x+4$$

Subtract x^{2} from both sides.

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

Subtract 4x from both sides.

$$4x-x^{2}-4x=4$$

Combine 4x and -4x to get 0.

$$-x^{2}=4$$

Divide both sides by -1.

$$x^{2}=-4$$

The equation is now solved.

$$x=2i x=-2i$$

Substitute 2i for x in the equation 2\sqrt{x}-x-2=0.

$$2\sqrt{2i}-2i-2=0$$

Simplify. The value x=2i satisfies the equation.

$$0=0$$

Substitute -2i for x in the equation 2\sqrt{x}-x-2=0.

$$2\sqrt{-2i}-\left(-2i\right)-2=0$$

Simplify. The value x=-2i satisfies the equation.

$$0=0$$

List all solutions of 2\sqrt{x}=x+2.

$$x=2i x=-2i$$