Solve 4x^2+2x/3=0

$$4 { x }^{ 2 } + \frac{ 2x }{ 3 } =0$$

Solve for x

$$x=-\frac{1}{6}\approx -0.166666667$$

$$x=0$$

Steps Using Factoring

Multiply both sides of the equation by 3.

$$12x^{2}+2x=0$$

Factor out x.

$$x\left(12x+2\right)=0$$

To find equation solutions, solve x=0 and 12x+2=0.

$$x=0 x=-\frac{1}{6}$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 3.

$$12x^{2}+2x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 12 for a, 2 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2±\sqrt{2^{2}}}{2\times 12}$$

Take the square root of 2^{2}.

$$x=\frac{-2±2}{2\times 12}$$

Multiply 2 times 12.

$$x=\frac{-2±2}{24}$$

Now solve the equation x=\frac{-2±2}{24} when ± is plus. Add -2 to 2.

$$x=\frac{0}{24}$$

Divide 0 by 24.

$$x=0$$

Now solve the equation x=\frac{-2±2}{24} when ± is minus. Subtract 2 from -2.

$$x=-\frac{4}{24}$$

Reduce the fraction \frac{-4}{24} to lowest terms by extracting and canceling out 4.

$$x=-\frac{1}{6}$$

The equation is now solved.

$$x=0 x=-\frac{1}{6}$$

Steps for Completing the Square

Multiply both sides of the equation by 3.

$$12x^{2}+2x=0$$

Divide both sides by 12.

$$\frac{12x^{2}+2x}{12}=\frac{0}{12}$$

Dividing by 12 undoes the multiplication by 12.

$$x^{2}+\frac{2}{12}x=\frac{0}{12}$$

Reduce the fraction \frac{2}{12} to lowest terms by extracting and canceling out 2.

$$x^{2}+\frac{1}{6}x=\frac{0}{12}$$

Divide 0 by 12.

$$x^{2}+\frac{1}{6}x=0$$

Divide \frac{1}{6}, the coefficient of the x term, by 2 to get \frac{1}{12}. Then add the square of \frac{1}{12} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{1}{6}x+\left(\frac{1}{12}\right)^{2}=\left(\frac{1}{12}\right)^{2}$$

Square \frac{1}{12} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{1}{6}x+\frac{1}{144}=\frac{1}{144}$$

Factor x^{2}+\frac{1}{6}x+\frac{1}{144}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{1}{12}\right)^{2}=\frac{1}{144}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{12}\right)^{2}}=\sqrt{\frac{1}{144}}$$

Simplify.

$$x+\frac{1}{12}=\frac{1}{12} x+\frac{1}{12}=-\frac{1}{12}$$

Subtract \frac{1}{12} from both sides of the equation.

$$x=0 x=-\frac{1}{6}$$