Solve 4x+3x(2x)=0

$$4 x + 3 x ( 2 x ) = 0$$

Solve for x

$$x=-\frac{2}{3}\approx -0.666666667$$

$$x=0$$

Steps Using Factoring

Factor out x.

$$x\left(4+2\times 3x\right)=0$$

To find equation solutions, solve x=0 and 4+6x=0.

$$x=0 x=-\frac{2}{3}$$

Steps Using the Quadratic Formula

Multiply x and x to get x^{2}.

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

Multiply 3 and 2 to get 6.

$$4x+6x^{2}=0$$

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.

$$6x^{2}+4x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 6 for a, 4 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-4±\sqrt{4^{2}}}{2\times 6}$$

Take the square root of 4^{2}.

$$x=\frac{-4±4}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{-4±4}{12}$$

Now solve the equation x=\frac{-4±4}{12} when ± is plus. Add -4 to 4.

$$x=\frac{0}{12}$$

Divide 0 by 12.

$$x=0$$

Now solve the equation x=\frac{-4±4}{12} when ± is minus. Subtract 4 from -4.

$$x=-\frac{8}{12}$$

Reduce the fraction \frac{-8}{12} to lowest terms by extracting and canceling out 4.

$$x=-\frac{2}{3}$$

The equation is now solved.

$$x=0 x=-\frac{2}{3}$$

Steps for Completing the Square

Multiply x and x to get x^{2}.

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

Multiply 3 and 2 to get 6.

$$4x+6x^{2}=0$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$6x^{2}+4x=0$$

Divide both sides by 6.

$$\frac{6x^{2}+4x}{6}=\frac{0}{6}$$

Dividing by 6 undoes the multiplication by 6.

$$x^{2}+\frac{4}{6}x=\frac{0}{6}$$

Reduce the fraction \frac{4}{6} to lowest terms by extracting and canceling out 2.

$$x^{2}+\frac{2}{3}x=\frac{0}{6}$$

Divide 0 by 6.

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

Divide \frac{2}{3}, the coefficient of the x term, by 2 to get \frac{1}{3}. Then add the square of \frac{1}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{2}{3}x+\left(\frac{1}{3}\right)^{2}=\left(\frac{1}{3}\right)^{2}$$

Square \frac{1}{3} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{2}{3}x+\frac{1}{9}=\frac{1}{9}$$

Factor x^{2}+\frac{2}{3}x+\frac{1}{9}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{1}{3}\right)^{2}=\frac{1}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{3}\right)^{2}}=\sqrt{\frac{1}{9}}$$

Simplify.

$$x+\frac{1}{3}=\frac{1}{3} x+\frac{1}{3}=-\frac{1}{3}$$

Subtract \frac{1}{3} from both sides of the equation.

$$x=0 x=-\frac{2}{3}$$