When factoring quadratic expressions, we are rewriting a quadratic from the form into the form (Note: the variables a and b are not necessarily equivalent in both forms: they just represent a variable number).
To better understand what we are looking for when we are factoring a quadratic, it is helpful to look at how we go in the opposite direction. This process is known as binomial multiplication, and is modeled by FOIL: First, Outside, Inside, Last.
Look at the coefficient of the x-term, and the constant term. When factoring a quadratic expression, we are looking for two integers, p and q. The sum of p and q is the coefficient of the x-term, and the product of p and q is the constant term.
In the example above, we can factor x2 – x – 6 as (x + 2)(x – 3). We know this is true, because we FOILed (x + 2)(x – 3) to get x2 – x – 6. Thinking about the relationship between factoring and FOIL, we see that the sum of 2 and –3 is –1 (the coefficient of x), and the product of 2 and -3 is -6 (the constant term).
Basic Quadratic Factoring
When factoring a quadratic, we need to identify two integers whose product is the constant term of the quadratic, and whose sum is the x-term coefficient. To identify these two integers, we identify pairs of factors that multiply to the constant term. From there, we add the integers in each pair together, and work with the pair that sums to the x-term coefficient.
To begin, we will list pairs of integers that multiply to 15:
Note that it can include two negative numbers, because the product will be positive. Next, we add p and q, looking for a sum of 8. Once we find our sum of 8, there is no point in finding the other sums, because we know we don't want to use those values for p and q.
We have identified p as 3, and q as 5. This means that we can factor the quadratic as:
The Sign of pq
Looking at the sign of the constant term can help you eliminate possible values for p and q when factoring. For example, if the constant term is negative, we know that one of p or q must be negative, but not both (because two negative values result in a positive number when multiplied). The table below can guide you when thinking about the sign of pq:
Here is how we can use the sign of pq to narrow our focus when factoring:
We see that the constant term is positive. This means that the signs of p and q must match: either both of them are positive numbers, or both of them are negative. Next, we look at the x-term. The coefficient is a negative number. This means that the sum of p and q must be negative. Since the sum of two positive numbers is always positive, we can conclude that p and q must be negative.
So we draft up pairs of negative numbers that equal positive 12 when multiplied:
Note that there is no need to write the other pairs of p and q that essentially switch what we already have. Their sums will be the same, because addition is commutative.
Now we add our integer pairs, and stop when we reach –7:
This means we can factor as: