Solve v^2=-19v

$$v ^ { 2 } = - 19 v$$

Solve for v

$$v=-19$$

$$v=0$$

Steps Using Factoring

Add 19v to both sides.

$$v^{2}+19v=0$$

Factor out v.

$$v\left(v+19\right)=0$$

To find equation solutions, solve v=0 and v+19=0.

$$v=0 v=-19$$

Steps Using the Quadratic Formula

Add 19v to both sides.

$$v^{2}+19v=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 19 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$v=\frac{-19±\sqrt{19^{2}}}{2}$$

Take the square root of 19^{2}.

$$v=\frac{-19±19}{2}$$

Now solve the equation v=\frac{-19±19}{2} when ± is plus. Add -19 to 19.

$$v=\frac{0}{2}$$

Divide 0 by 2.

$$v=0$$

Now solve the equation v=\frac{-19±19}{2} when ± is minus. Subtract 19 from -19.

$$v=-\frac{38}{2}$$

Divide -38 by 2.

$$v=-19$$

The equation is now solved.

$$v=0 v=-19$$

Steps for Completing the Square

Add 19v to both sides.

$$v^{2}+19v=0$$

Divide 19, the coefficient of the x term, by 2 to get \frac{19}{2}. Then add the square of \frac{19}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$v^{2}+19v+\left(\frac{19}{2}\right)^{2}=\left(\frac{19}{2}\right)^{2}$$

Square \frac{19}{2} by squaring both the numerator and the denominator of the fraction.

$$v^{2}+19v+\frac{361}{4}=\frac{361}{4}$$

Factor v^{2}+19v+\frac{361}{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}.

$$\left(v+\frac{19}{2}\right)^{2}=\frac{361}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(v+\frac{19}{2}\right)^{2}}=\sqrt{\frac{361}{4}}$$

Simplify.

$$v+\frac{19}{2}=\frac{19}{2} v+\frac{19}{2}=-\frac{19}{2}$$

Subtract \frac{19}{2} from both sides of the equation.

$$v=0 v=-19$$