Solve v^2+3v+7v+21

$$v ^ { 2 } + 3 v + 7 v + 21$$

Factor

$$\left(v+3\right)\left(v+7\right)$$

Solution Steps

Multiply and combine like terms.

$$v^{2}+10v+21$$

Factor the expression by grouping. First, the expression needs to be rewritten as v^{2}+av+bv+21. To find a and b, set up a system to be solved.

$$a+b=10 ab=1\times 21=21$$

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 21.

$$1,21 3,7$$

Calculate the sum for each pair.

$$1+21=22 3+7=10$$

The solution is the pair that gives sum 10.

$$a=3 b=7$$

Rewrite v^{2}+10v+21 as \left(v^{2}+3v\right)+\left(7v+21\right).

$$\left(v^{2}+3v\right)+\left(7v+21\right)$$

Factor out v in the first and 7 in the second group.

$$v\left(v+3\right)+7\left(v+3\right)$$

Factor out common term v+3 by using distributive property.

$$\left(v+3\right)\left(v+7\right)$$

Evaluate

$$\left(v+3\right)\left(v+7\right)$$

Solution Steps

Combine 3v and 7v to get 10v.

$$v^{2}+10v+21$$