Q. 149E - Intermediate Algebra

Question

In the following exercises, factor each expression using any method $24n^2+20n+4$

Step-by-Step Solution

Answer

The factors are $4\left(2n+1\right)\left(3n+1\right)$

Step 1. Given Information

The trinomial is $24n^2+20n+4$

Find factors using 'ac' method. 

Step 2. Find GCF

Find GCF and factor it

GCF is 4

So, trinomial becomes $4\left(6n^2+5n+1\right)$

Step 3. Find 'ac'

Compare the trinomial inside the parenthesis with $a{x}^2+bx+c$

So, $a=6, b=5, c=1$

so, $ac=6$

Find two number such as their product is  6 and sum is 5

So numbers are 2,3.

Step 4. Split the middle and factor trinomial by grouping

Split the middle term 

$4\left(6n^2+3n+2n+1\right)$

Factor it

$$\begin{gathered} 4\left(3n\left(2n+1\right)+1\left(2n+1\right)\right) \\ 4\left(2n+1\right)\left(3n+1\right) \end{gathered}$$

Step 5. Check

Multiply the factors $4\left(2n+1\right)\left(3n+1\right)$

$$\begin{gathered} 4\left(6n^2+2n+3n+1\right) \\ 4\left(6n^2+5n+1\right) \\ 24n^2+20n+4 \end{gathered}$$