Solve v^2+5v+6v+30

$$v ^ { 2 } + 5 v + 6 v + 30$$

Factor

$$\left(v+5\right)\left(v+6\right)$$

Solution Steps

Multiply and combine like terms.

$$v^{2}+11v+30$$

Factor the expression by grouping. First, the expression needs to be rewritten as v^{2}+av+bv+30. To find a and b, set up a system to be solved.

$$a+b=11 ab=1\times 30=30$$

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 30.

$$1,30 2,15 3,10 5,6$$

Calculate the sum for each pair.

$$1+30=31 2+15=17 3+10=13 5+6=11$$

The solution is the pair that gives sum 11.

$$a=5 b=6$$

Rewrite v^{2}+11v+30 as \left(v^{2}+5v\right)+\left(6v+30\right).

$$\left(v^{2}+5v\right)+\left(6v+30\right)$$

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

$$v\left(v+5\right)+6\left(v+5\right)$$

Factor out common term v+5 by using distributive property.

$$\left(v+5\right)\left(v+6\right)$$

Evaluate

$$\left(v+5\right)\left(v+6\right)$$

Solution Steps

Combine 5v and 6v to get 11v.

$$v^{2}+11v+30$$