Solve 6n^2-7n

$$6 n ^ { 2 } - 7 n$$

Factor

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

Solution Steps

Factor out n.

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

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.

$$6n^{2}-7n=0$$

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.

$$n=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}}}{2\times 6}$$

Take the square root of \left(-7\right)^{2}.

$$n=\frac{-\left(-7\right)±7}{2\times 6}$$

The opposite of -7 is 7.

$$n=\frac{7±7}{2\times 6}$$

Multiply 2 times 6.

$$n=\frac{7±7}{12}$$

Now solve the equation n=\frac{7±7}{12} when ± is plus. Add 7 to 7.

$$n=\frac{14}{12}$$

Reduce the fraction \frac{14}{12} to lowest terms by extracting and canceling out 2.

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

Now solve the equation n=\frac{7±7}{12} when ± is minus. Subtract 7 from 7.

$$n=\frac{0}{12}$$

Divide 0 by 12.

$$n=0$$

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

$$6n^{2}-7n=6\left(n-\frac{7}{6}\right)n$$

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

$$6n^{2}-7n=6\times \frac{6n-7}{6}n$$

Cancel out 6, the greatest common factor in 6 and 6.

$$6n^{2}-7n=\left(6n-7\right)n$$

Evaluate

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