Solve 1/2x^2+9x+36

$$\frac { 1 } { 2 } x ^ { 2 } + 9 x + 36$$

Factor

$$\frac{\left(x+6\right)\left(x+12\right)}{2}$$

Solution Steps

Factor out \frac{1}{2}.

$$\frac{x^{2}+18x+72}{2}$$

Consider x^{2}+18x+72. Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+72. To find a and b, set up a system to be solved.

$$a+b=18 ab=1\times 72=72$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 72.

$$1,72 2,36 3,24 4,18 6,12 8,9$$

Calculate the sum for each pair.

$$1+72=73 2+36=38 3+24=27 4+18=22 6+12=18 8+9=17$$

The solution is the pair that gives sum 18.

$$a=6 b=12$$

Rewrite x^{2}+18x+72 as \left(x^{2}+6x\right)+\left(12x+72\right).

$$\left(x^{2}+6x\right)+\left(12x+72\right)$$

Factor out x in the first and 12 in the second group.

$$x\left(x+6\right)+12\left(x+6\right)$$

Factor out common term x+6 by using distributive property.

$$\left(x+6\right)\left(x+12\right)$$

Rewrite the complete factored expression.

$$\frac{\left(x+6\right)\left(x+12\right)}{2}$$

Evaluate

$$\frac{\left(x+6\right)\left(x+12\right)}{2}$$