Solve 18v-21v^2

$$18 v - 21 v ^ { 2 }$$

Factor

$$3v\left(6-7v\right)$$

Solution Steps

Factor out 3.

$$3\left(6v-7v^{2}\right)$$

Consider 6v-7v^{2}. Factor out v.

$$v\left(6-7v\right)$$

Rewrite the complete factored expression.

$$3v\left(-7v+6\right)$$

Steps Using the Quadratic Formula

Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.

$$-21v^{2}+18v=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$v=\frac{-18±\sqrt{18^{2}}}{2\left(-21\right)}$$

Take the square root of 18^{2}.

$$v=\frac{-18±18}{2\left(-21\right)}$$

Multiply 2 times -21.

$$v=\frac{-18±18}{-42}$$

Now solve the equation v=\frac{-18±18}{-42} when ± is plus. Add -18 to 18.

$$v=\frac{0}{-42}$$

Divide 0 by -42.

$$v=0$$

Now solve the equation v=\frac{-18±18}{-42} when ± is minus. Subtract 18 from -18.

$$v=-\frac{36}{-42}$$

Reduce the fraction \frac{-36}{-42} to lowest terms by extracting and canceling out 6.

$$v=\frac{6}{7}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 0 for x_{1} and \frac{6}{7} for x_{2}.

$$-21v^{2}+18v=-21v\left(v-\frac{6}{7}\right)$$

Subtract \frac{6}{7} from v by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$-21v^{2}+18v=-21v\times \frac{-7v+6}{-7}$$

Cancel out 7, the greatest common factor in -21 and -7.

$$-21v^{2}+18v=3v\left(-7v+6\right)$$

Evaluate

$$3v\left(6-7v\right)$$