Solve 15n^2-5n

$$15 n ^ { 2 } - 5 n$$

Factor

$$5n\left(3n-1\right)$$

Solution Steps

Factor out 5.

$$5\left(3n^{2}-n\right)$$

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

$$n\left(3n-1\right)$$

Rewrite the complete factored expression.

$$5n\left(3n-1\right)$$

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.

$$15n^{2}-5n=0$$

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.

$$n=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}}}{2\times 15}$$

Take the square root of \left(-5\right)^{2}.

$$n=\frac{-\left(-5\right)±5}{2\times 15}$$

The opposite of -5 is 5.

$$n=\frac{5±5}{2\times 15}$$

Multiply 2 times 15.

$$n=\frac{5±5}{30}$$

Now solve the equation n=\frac{5±5}{30} when ± is plus. Add 5 to 5.

$$n=\frac{10}{30}$$

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

$$n=\frac{1}{3}$$

Now solve the equation n=\frac{5±5}{30} when ± is minus. Subtract 5 from 5.

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

Divide 0 by 30.

$$n=0$$

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

$$15n^{2}-5n=15\left(n-\frac{1}{3}\right)n$$

Subtract \frac{1}{3} from n by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$15n^{2}-5n=15\times \frac{3n-1}{3}n$$

Cancel out 3, the greatest common factor in 15 and 3.

$$15n^{2}-5n=5\left(3n-1\right)n$$

Evaluate

$$5n\left(3n-1\right)$$