Solve 18n^2-909n

$$18 n ^ { 2 } - 909 n$$

Factor

$$9n\left(2n-101\right)$$

Solution Steps

Factor out 9.

$$9\left(2n^{2}-101n\right)$$

Consider 2n^{2}-101n. Factor out n.

$$n\left(2n-101\right)$$

Rewrite the complete factored expression.

$$9n\left(2n-101\right)$$

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.

$$18n^{2}-909n=0$$

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.

$$n=\frac{-\left(-909\right)±\sqrt{\left(-909\right)^{2}}}{2\times 18}$$

Take the square root of \left(-909\right)^{2}.

$$n=\frac{-\left(-909\right)±909}{2\times 18}$$

The opposite of -909 is 909.

$$n=\frac{909±909}{2\times 18}$$

Multiply 2 times 18.

$$n=\frac{909±909}{36}$$

Now solve the equation n=\frac{909±909}{36} when ± is plus. Add 909 to 909.

$$n=\frac{1818}{36}$$

Reduce the fraction \frac{1818}{36} to lowest terms by extracting and canceling out 18.

$$n=\frac{101}{2}$$

Now solve the equation n=\frac{909±909}{36} when ± is minus. Subtract 909 from 909.

$$n=\frac{0}{36}$$

Divide 0 by 36.

$$n=0$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{101}{2} for x_{1} and 0 for x_{2}.

$$18n^{2}-909n=18\left(n-\frac{101}{2}\right)n$$

Subtract \frac{101}{2} from n by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$18n^{2}-909n=18\times \frac{2n-101}{2}n$$

Cancel out 2, the greatest common factor in 18 and 2.

$$18n^{2}-909n=9\left(2n-101\right)n$$

Evaluate

$$9n\left(2n-101\right)$$