Solve 1/2x^2=x

$$\frac{ 1 }{ 2 } { x }^{ 2 } = x$$

Solve for x

$$x=2$$

$$x=0$$

Steps Using Factoring

Subtract x from both sides.

$$\frac{1}{2}x^{2}-x=0$$

Factor out x.

$$x\left(\frac{1}{2}x-1\right)=0$$

To find equation solutions, solve x=0 and \frac{x}{2}-1=0.

$$x=0 x=2$$

Steps Using the Quadratic Formula

Subtract x from both sides.

$$\frac{1}{2}x^{2}-x=0$$

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

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

Take the square root of 1.

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

The opposite of -1 is 1.

$$x=\frac{1±1}{2\times \frac{1}{2}}$$

Multiply 2 times \frac{1}{2}.

$$x=\frac{1±1}{1}$$

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

$$x=\frac{2}{1}$$

Divide 2 by 1.

$$x=2$$

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

$$x=\frac{0}{1}$$

Divide 0 by 1.

$$x=0$$

The equation is now solved.

$$x=2 x=0$$

Steps for Completing the Square

Subtract x from both sides.

$$\frac{1}{2}x^{2}-x=0$$

Multiply both sides by 2.

$$\frac{\frac{1}{2}x^{2}-x}{\frac{1}{2}}=\frac{0}{\frac{1}{2}}$$

Dividing by \frac{1}{2} undoes the multiplication by \frac{1}{2}.

$$x^{2}+\left(-\frac{1}{\frac{1}{2}}\right)x=\frac{0}{\frac{1}{2}}$$

Divide -1 by \frac{1}{2} by multiplying -1 by the reciprocal of \frac{1}{2}.

$$x^{2}-2x=\frac{0}{\frac{1}{2}}$$

Divide 0 by \frac{1}{2} by multiplying 0 by the reciprocal of \frac{1}{2}.

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

Divide -2, the coefficient of the x term, by 2 to get -1. Then add the square of -1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-2x+1=1$$

Factor x^{2}-2x+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

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

Take the square root of both sides of the equation.

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

Simplify.

$$x-1=1 x-1=-1$$

Add 1 to both sides of the equation.

$$x=2 x=0$$