Solve y^2+13/2y+3

$$y ^ { 2 } + \frac { 13 } { 2 } y + 3$$

Factor

$$\frac{\left(y+6\right)\left(2y+1\right)}{2}$$

Solution Steps

Factor out \frac{1}{2}.

$$\frac{2y^{2}+13y+6}{2}$$

Consider 2y^{2}+13y+6. Factor the expression by grouping. First, the expression needs to be rewritten as 2y^{2}+ay+by+6. To find a and b, set up a system to be solved.

$$a+b=13 ab=2\times 6=12$$

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 12.

$$1,12 2,6 3,4$$

Calculate the sum for each pair.

$$1+12=13 2+6=8 3+4=7$$

The solution is the pair that gives sum 13.

$$a=1 b=12$$

Rewrite 2y^{2}+13y+6 as \left(2y^{2}+y\right)+\left(12y+6\right).

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

Factor out y in the first and 6 in the second group.

$$y\left(2y+1\right)+6\left(2y+1\right)$$

Factor out common term 2y+1 by using distributive property.

$$\left(2y+1\right)\left(y+6\right)$$

Rewrite the complete factored expression.

$$\frac{\left(2y+1\right)\left(y+6\right)}{2}$$

Evaluate

$$y^{2}+\frac{13y}{2}+3$$