Solve 9y^2-6y

$$9 y ^ { 2 } - 6 y$$

Factor

$$3y\left(3y-2\right)$$

Solution Steps

Factor out 3.

$$3\left(3y^{2}-2y\right)$$

Consider 3y^{2}-2y. Factor out y.

$$y\left(3y-2\right)$$

Rewrite the complete factored expression.

$$3y\left(3y-2\right)$$

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.

$$9y^{2}-6y=0$$

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.

$$y=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}}}{2\times 9}$$

Take the square root of \left(-6\right)^{2}.

$$y=\frac{-\left(-6\right)±6}{2\times 9}$$

The opposite of -6 is 6.

$$y=\frac{6±6}{2\times 9}$$

Multiply 2 times 9.

$$y=\frac{6±6}{18}$$

Now solve the equation y=\frac{6±6}{18} when ± is plus. Add 6 to 6.

$$y=\frac{12}{18}$$

Reduce the fraction \frac{12}{18} to lowest terms by extracting and canceling out 6.

$$y=\frac{2}{3}$$

Now solve the equation y=\frac{6±6}{18} when ± is minus. Subtract 6 from 6.

$$y=\frac{0}{18}$$

Divide 0 by 18.

$$y=0$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{2}{3} for x_{1} and 0 for x_{2}.

$$9y^{2}-6y=9\left(y-\frac{2}{3}\right)y$$

Subtract \frac{2}{3} from y by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$9y^{2}-6y=9\times \frac{3y-2}{3}y$$

Cancel out 3, the greatest common factor in 9 and 3.

$$9y^{2}-6y=3\left(3y-2\right)y$$

Evaluate

$$3y\left(3y-2\right)$$