Factoring Quadratic Trinomials: $ax^2 + bx + c$

In this lesson we will be factoring quadratic trinomial. The method we are going to use was discovered by Kenny’s class and we use it when the coefficient in front of $x^2$ is greater than 1. This method may seem a little strange at first but once you get the hang of it, you will find you can solve problems like this quite quickly. (To learn more about Kenny go to the About Us page.)

Steps for Kenny’s Method for factoring $ax^2 + bx + c$:

 
1. Is there a common factor in the trinomial? If so factor it out.
2. Multiply the coefficients of the first and last terms $a * b$
3. List all the factors of $a * c$ and determine which of these, when added, would have given us a middle term $b$
4. Next we will need to find the greatest common factor of each row and column.
5. Always check your work.
For example in the last example in the video if we forgot to take out the greatest common factor first and tried to factor $10x^2 – 34x + 12$ we would have gotten $(10x – 4)(2x – 3)$ and when we check that we would have gotten $20x^2 – 38x – 12$. So if the problem does not check, go back and make sure you first factored out all the common factors.
 

Your turn

Try to factor:   $6x^2$ – $15x$ – $36$.
 
Solution using Kenny’s Method:

1. Check if there are any common factors in $6x^2$ – $15x$ – $36$.
Since $3$ is a factor of $6$, $15$ and $36$ we can factor this out, leaving us with:
$3(2x^2$ – $15x$ – $12)$

2. We will start off as if we are multiplying the two factor using the box method. The key to factoring $(2x^2 – 15x – 12)$ is to look at the first and last terms.

3. Next we multiply the coefficients of the first and last terms,
$2 * 12 = 24$.

4.  We need to list all the factors of $24$ and determine which of these would have given us a middle term  $-5x$.  Since the middle term is negative and the last term is negative, we know that the one of the factors of $24$ will be negative and the other will be positive.  When we find the sum that equals $-5$, we will have found the terms that we are looking for.

5. Next we will put $-8x$ and $3x$ into the remaining two squares, since they add to $-5x$.

6. Next we will need to find the greatest common factor of each row and column.

The GCF of	The GCF of	The GCF of	The GCF of
$2x^2$ and $-8x$	$3x$ and $-12$	$2x^2$ and $3x$	$-8x$ and $-12$
is $2x$	is $3$	is $x$	is $-4$

7. So the factors of $6x^2 – 15x – 36$ are $3(2x + 3)(x – 4)$.

8. Always check your work.
When we multiply $3(2x + 3)(x – 4)$ we do get $6x^2 – 15x – 36$, but if the problem does not check, go back and make sure you first factored out all the common factors.