Solve sqrt{-x}=x

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

Solve for x

$$x=0$$

Solution Steps

Square both sides of the equation.

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

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

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

Subtract x^{2} from both sides.

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

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

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

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -1 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1\right)±\sqrt{1}}{2\left(-1\right)}$$

Take the square root of 1.

$$x=\frac{-\left(-1\right)±1}{2\left(-1\right)}$$

The opposite of -1 is 1.

$$x=\frac{1±1}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{1±1}{-2}$$

Now solve the equation x=\frac{1±1}{-2} when ± is plus. Add 1 to 1.

$$x=\frac{2}{-2}$$

Divide 2 by -2.

$$x=-1$$

Now solve the equation x=\frac{1±1}{-2} when ± is minus. Subtract 1 from 1.

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

Divide 0 by -2.

$$x=0$$

The equation is now solved.

$$x=-1 x=0$$

Substitute -1 for x in the equation \sqrt{-x}=x.

$$\sqrt{-\left(-1\right)}=-1$$

Simplify. The value x=-1 does not satisfy the equation because the left and the right hand side have opposite signs.

$$1=-1$$

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

$$\sqrt{0}=0$$

Simplify. The value x=0 satisfies the equation.

$$0=0$$

Equation \sqrt{-x}=x has a unique solution.

$$x=0$$