Solve v^2=11v

$$v ^ { 2 } = 11 v$$

Solve for v

$$v=11$$

$$v=0$$

Steps Using Factoring

Subtract 11v from both sides.

$$v^{2}-11v=0$$

Factor out v.

$$v\left(v-11\right)=0$$

To find equation solutions, solve v=0 and v-11=0.

$$v=0 v=11$$

Steps Using the Quadratic Formula

Subtract 11v from both sides.

$$v^{2}-11v=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -11 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$v=\frac{-\left(-11\right)±\sqrt{\left(-11\right)^{2}}}{2}$$

Take the square root of \left(-11\right)^{2}.

$$v=\frac{-\left(-11\right)±11}{2}$$

The opposite of -11 is 11.

$$v=\frac{11±11}{2}$$

Now solve the equation v=\frac{11±11}{2} when ± is plus. Add 11 to 11.

$$v=\frac{22}{2}$$

Divide 22 by 2.

$$v=11$$

Now solve the equation v=\frac{11±11}{2} when ± is minus. Subtract 11 from 11.

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

Divide 0 by 2.

$$v=0$$

The equation is now solved.

$$v=11 v=0$$

Steps for Completing the Square

Subtract 11v from both sides.

$$v^{2}-11v=0$$

Divide -11, the coefficient of the x term, by 2 to get -\frac{11}{2}. Then add the square of -\frac{11}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$v^{2}-11v+\left(-\frac{11}{2}\right)^{2}=\left(-\frac{11}{2}\right)^{2}$$

Square -\frac{11}{2} by squaring both the numerator and the denominator of the fraction.

$$v^{2}-11v+\frac{121}{4}=\frac{121}{4}$$

Factor v^{2}-11v+\frac{121}{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}.

$$\left(v-\frac{11}{2}\right)^{2}=\frac{121}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(v-\frac{11}{2}\right)^{2}}=\sqrt{\frac{121}{4}}$$

Simplify.

$$v-\frac{11}{2}=\frac{11}{2} v-\frac{11}{2}=-\frac{11}{2}$$

Add \frac{11}{2} to both sides of the equation.

$$v=11 v=0$$