Solve 1/2x^2+x+x=0

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

Solve for x

$$x=-4$$

$$x=0$$

Steps Using Factoring

Factor out x.

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

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

$$x=0 x=-4$$

Steps Using the Quadratic Formula

Combine x and x to get 2x.

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

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

$$x=\frac{-2±\sqrt{2^{2}}}{2\times \frac{1}{2}}$$

Take the square root of 2^{2}.

$$x=\frac{-2±2}{2\times \frac{1}{2}}$$

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

$$x=\frac{-2±2}{1}$$

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

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

Divide 0 by 1.

$$x=0$$

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

$$x=-\frac{4}{1}$$

Divide -4 by 1.

$$x=-4$$

The equation is now solved.

$$x=0 x=-4$$

Steps for Completing the Square

Combine x and x to get 2x.

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

Multiply both sides by 2.

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

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

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

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

$$x^{2}+4x=\frac{0}{\frac{1}{2}}$$

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

$$x^{2}+4x=0$$

Divide 4, the coefficient of the x term, by 2 to get 2. Then add the square of 2 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square 2.

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

Factor x^{2}+4x+4. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

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

Take the square root of both sides of the equation.

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

Simplify.

$$x+2=2 x+2=-2$$

Subtract 2 from both sides of the equation.

$$x=0 x=-4$$