Solve 5v(3v-5)=0

$$5 v ( 3 v - 5 ) = 0$$

Solve for v

$$v = \frac{5}{3} = 1\frac{2}{3} \approx 1.666666667$$

$$v=0$$

Steps Using Factoring

Use the distributive property to multiply 5v by 3v-5.

$$15v^{2}-25v=0$$

Factor out v.

$$v\left(15v-25\right)=0$$

To find equation solutions, solve v=0 and 15v-25=0.

$$v=0 v=\frac{5}{3}$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 5v by 3v-5.

$$15v^{2}-25v=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 15 for a, -25 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$v=\frac{-\left(-25\right)±\sqrt{\left(-25\right)^{2}}}{2\times 15}$$

Take the square root of \left(-25\right)^{2}.

$$v=\frac{-\left(-25\right)±25}{2\times 15}$$

The opposite of -25 is 25.

$$v=\frac{25±25}{2\times 15}$$

Multiply 2 times 15.

$$v=\frac{25±25}{30}$$

Now solve the equation v=\frac{25±25}{30} when ± is plus. Add 25 to 25.

$$v=\frac{50}{30}$$

Reduce the fraction \frac{50}{30} to lowest terms by extracting and canceling out 10.

$$v=\frac{5}{3}$$

Now solve the equation v=\frac{25±25}{30} when ± is minus. Subtract 25 from 25.

$$v=\frac{0}{30}$$

Divide 0 by 30.

$$v=0$$

The equation is now solved.

$$v=\frac{5}{3} v=0$$

Steps for Completing the Square

Use the distributive property to multiply 5v by 3v-5.

$$15v^{2}-25v=0$$

Divide both sides by 15.

$$\frac{15v^{2}-25v}{15}=\frac{0}{15}$$

Dividing by 15 undoes the multiplication by 15.

$$v^{2}+\left(-\frac{25}{15}\right)v=\frac{0}{15}$$

Reduce the fraction \frac{-25}{15} to lowest terms by extracting and canceling out 5.

$$v^{2}-\frac{5}{3}v=\frac{0}{15}$$

Divide 0 by 15.

$$v^{2}-\frac{5}{3}v=0$$

Divide -\frac{5}{3}, the coefficient of the x term, by 2 to get -\frac{5}{6}. Then add the square of -\frac{5}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$v^{2}-\frac{5}{3}v+\left(-\frac{5}{6}\right)^{2}=\left(-\frac{5}{6}\right)^{2}$$

Square -\frac{5}{6} by squaring both the numerator and the denominator of the fraction.

$$v^{2}-\frac{5}{3}v+\frac{25}{36}=\frac{25}{36}$$

Factor v^{2}-\frac{5}{3}v+\frac{25}{36}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(v-\frac{5}{6}\right)^{2}=\frac{25}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(v-\frac{5}{6}\right)^{2}}=\sqrt{\frac{25}{36}}$$

Simplify.

$$v-\frac{5}{6}=\frac{5}{6} v-\frac{5}{6}=-\frac{5}{6}$$

Add \frac{5}{6} to both sides of the equation.

$$v=\frac{5}{3} v=0$$