Solve 22v^2=-21v

$$22 v ^ { 2 } = - 21 v$$

Solve for v

$$v=-\frac{21}{22}\approx -0.954545455$$

$$v=0$$

Steps Using Factoring

Add 21v to both sides.

$$22v^{2}+21v=0$$

Factor out v.

$$v\left(22v+21\right)=0$$

To find equation solutions, solve v=0 and 22v+21=0.

$$v=0 v=-\frac{21}{22}$$

Steps Using the Quadratic Formula

Add 21v to both sides.

$$22v^{2}+21v=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 22 for a, 21 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$v=\frac{-21±\sqrt{21^{2}}}{2\times 22}$$

Take the square root of 21^{2}.

$$v=\frac{-21±21}{2\times 22}$$

Multiply 2 times 22.

$$v=\frac{-21±21}{44}$$

Now solve the equation v=\frac{-21±21}{44} when ± is plus. Add -21 to 21.

$$v=\frac{0}{44}$$

Divide 0 by 44.

$$v=0$$

Now solve the equation v=\frac{-21±21}{44} when ± is minus. Subtract 21 from -21.

$$v=-\frac{42}{44}$$

Reduce the fraction \frac{-42}{44} to lowest terms by extracting and canceling out 2.

$$v=-\frac{21}{22}$$

The equation is now solved.

$$v=0 v=-\frac{21}{22}$$

Steps for Completing the Square

Add 21v to both sides.

$$22v^{2}+21v=0$$

Divide both sides by 22.

$$\frac{22v^{2}+21v}{22}=\frac{0}{22}$$

Dividing by 22 undoes the multiplication by 22.

$$v^{2}+\frac{21}{22}v=\frac{0}{22}$$

Divide 0 by 22.

$$v^{2}+\frac{21}{22}v=0$$

Divide \frac{21}{22}, the coefficient of the x term, by 2 to get \frac{21}{44}. Then add the square of \frac{21}{44} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$v^{2}+\frac{21}{22}v+\left(\frac{21}{44}\right)^{2}=\left(\frac{21}{44}\right)^{2}$$

Square \frac{21}{44} by squaring both the numerator and the denominator of the fraction.

$$v^{2}+\frac{21}{22}v+\frac{441}{1936}=\frac{441}{1936}$$

Factor v^{2}+\frac{21}{22}v+\frac{441}{1936}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(v+\frac{21}{44}\right)^{2}=\frac{441}{1936}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(v+\frac{21}{44}\right)^{2}}=\sqrt{\frac{441}{1936}}$$

Simplify.

$$v+\frac{21}{44}=\frac{21}{44} v+\frac{21}{44}=-\frac{21}{44}$$

Subtract \frac{21}{44} from both sides of the equation.

$$v=0 v=-\frac{21}{22}$$