Solve n^2+9n+6n+54

$$n ^ { 2 } + 9 n + 6 n + 54$$

Factor

$$\left(n+6\right)\left(n+9\right)$$

Solution Steps

Multiply and combine like terms.

$$n^{2}+15n+54$$

Factor the expression by grouping. First, the expression needs to be rewritten as n^{2}+an+bn+54. To find a and b, set up a system to be solved.

$$a+b=15 ab=1\times 54=54$$

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 54.

$$1,54 2,27 3,18 6,9$$

Calculate the sum for each pair.

$$1+54=55 2+27=29 3+18=21 6+9=15$$

The solution is the pair that gives sum 15.

$$a=6 b=9$$

Rewrite n^{2}+15n+54 as \left(n^{2}+6n\right)+\left(9n+54\right).

$$\left(n^{2}+6n\right)+\left(9n+54\right)$$

Factor out n in the first and 9 in the second group.

$$n\left(n+6\right)+9\left(n+6\right)$$

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

$$\left(n+6\right)\left(n+9\right)$$

Evaluate

$$\left(n+6\right)\left(n+9\right)$$

Solution Steps

Combine 9n and 6n to get 15n.

$$n^{2}+15n+54$$