Solve 4v^2=-12v

$$4 v ^ { 2 } = - 12 v$$

Solve for v

$$v=-3$$

$$v=0$$

Steps Using Factoring

Add 12v to both sides.

$$4v^{2}+12v=0$$

Factor out v.

$$v\left(4v+12\right)=0$$

To find equation solutions, solve v=0 and 4v+12=0.

$$v=0 v=-3$$

Steps Using the Quadratic Formula

Add 12v to both sides.

$$4v^{2}+12v=0$$

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

$$v=\frac{-12±\sqrt{12^{2}}}{2\times 4}$$

Take the square root of 12^{2}.

$$v=\frac{-12±12}{2\times 4}$$

Multiply 2 times 4.

$$v=\frac{-12±12}{8}$$

Now solve the equation v=\frac{-12±12}{8} when ± is plus. Add -12 to 12.

$$v=\frac{0}{8}$$

Divide 0 by 8.

$$v=0$$

Now solve the equation v=\frac{-12±12}{8} when ± is minus. Subtract 12 from -12.

$$v=-\frac{24}{8}$$

Divide -24 by 8.

$$v=-3$$

The equation is now solved.

$$v=0 v=-3$$

Steps for Completing the Square

Add 12v to both sides.

$$4v^{2}+12v=0$$

Divide both sides by 4.

$$\frac{4v^{2}+12v}{4}=\frac{0}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$v^{2}+\frac{12}{4}v=\frac{0}{4}$$

Divide 12 by 4.

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

Divide 0 by 4.

$$v^{2}+3v=0$$

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

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

Square \frac{3}{2} by squaring both the numerator and the denominator of the fraction.

$$v^{2}+3v+\frac{9}{4}=\frac{9}{4}$$

Factor v^{2}+3v+\frac{9}{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}.

$$\left(v+\frac{3}{2}\right)^{2}=\frac{9}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(v+\frac{3}{2}\right)^{2}}=\sqrt{\frac{9}{4}}$$

Simplify.

$$v+\frac{3}{2}=\frac{3}{2} v+\frac{3}{2}=-\frac{3}{2}$$

Subtract \frac{3}{2} from both sides of the equation.

$$v=0 v=-3$$