Solve -3/10n^2+11/10n+2

$$\frac { - 3 } { 10 } n ^ { 2 } + \frac { 11 } { 10 } n + 2$$

Factor

$$\frac{\left(5-n\right)\left(3n+4\right)}{10}$$

Solution Steps

Factor out \frac{1}{10}.

$$\frac{-3n^{2}+11n+20}{10}$$

Consider -3n^{2}+11n+20. Factor the expression by grouping. First, the expression needs to be rewritten as -3n^{2}+an+bn+20. To find a and b, set up a system to be solved.

$$a+b=11 ab=-3\times 20=-60$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -60.

$$-1,60 -2,30 -3,20 -4,15 -5,12 -6,10$$

Calculate the sum for each pair.

$$-1+60=59 -2+30=28 -3+20=17 -4+15=11 -5+12=7 -6+10=4$$

The solution is the pair that gives sum 11.

$$a=15 b=-4$$

Rewrite -3n^{2}+11n+20 as \left(-3n^{2}+15n\right)+\left(-4n+20\right).

$$\left(-3n^{2}+15n\right)+\left(-4n+20\right)$$

Factor out 3n in the first and 4 in the second group.

$$3n\left(-n+5\right)+4\left(-n+5\right)$$

Factor out common term -n+5 by using distributive property.

$$\left(-n+5\right)\left(3n+4\right)$$

Rewrite the complete factored expression.

$$\frac{\left(-n+5\right)\left(3n+4\right)}{10}$$

Evaluate

$$-\frac{3n^{2}}{10}+\frac{11n}{10}+2$$

Solution Steps

Fraction \frac{-3}{10} can be rewritten as -\frac{3}{10} by extracting the negative sign.

$$-\frac{3}{10}n^{2}+\frac{11}{10}n+2$$