Solve 1/2x^2+9x+28

$$\frac { 1 } { 2 } x ^ { 2 } + 9 x + 28$$

Factor

$$\frac{\left(x+4\right)\left(x+14\right)}{2}$$

Solution Steps

Factor out \frac{1}{2}.

$$\frac{x^{2}+18x+56}{2}$$

Consider x^{2}+18x+56. Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+56. To find a and b, set up a system to be solved.

$$a+b=18 ab=1\times 56=56$$

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 56.

$$1,56 2,28 4,14 7,8$$

Calculate the sum for each pair.

$$1+56=57 2+28=30 4+14=18 7+8=15$$

The solution is the pair that gives sum 18.

$$a=4 b=14$$

Rewrite x^{2}+18x+56 as \left(x^{2}+4x\right)+\left(14x+56\right).

$$\left(x^{2}+4x\right)+\left(14x+56\right)$$

Factor out x in the first and 14 in the second group.

$$x\left(x+4\right)+14\left(x+4\right)$$

Factor out common term x+4 by using distributive property.

$$\left(x+4\right)\left(x+14\right)$$

Rewrite the complete factored expression.

$$\frac{\left(x+4\right)\left(x+14\right)}{2}$$

Evaluate

$$\frac{\left(x+4\right)\left(x+14\right)}{2}$$