Solve 3z^2-2z

$$3 z ^ { 2 } - 2 z$$

Factor

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

Solution Steps

Factor out z.

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

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.

$$3z^{2}-2z=0$$

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.

$$z=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}}}{2\times 3}$$

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

$$z=\frac{-\left(-2\right)±2}{2\times 3}$$

The opposite of -2 is 2.

$$z=\frac{2±2}{2\times 3}$$

Multiply 2 times 3.

$$z=\frac{2±2}{6}$$

Now solve the equation z=\frac{2±2}{6} when ± is plus. Add 2 to 2.

$$z=\frac{4}{6}$$

Reduce the fraction \frac{4}{6} to lowest terms by extracting and canceling out 2.

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

Now solve the equation z=\frac{2±2}{6} when ± is minus. Subtract 2 from 2.

$$z=\frac{0}{6}$$

Divide 0 by 6.

$$z=0$$

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}.

$$3z^{2}-2z=3\left(z-\frac{2}{3}\right)z$$

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

$$3z^{2}-2z=3\times \frac{3z-2}{3}z$$

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

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

Evaluate

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