Solving Quadratic Equations by Factoring
Before we begin our discussion of solving quadratic equations by factoring, let us review the following important mathematical fact:
If , then either , or both equal zero.
The process of solving quadratic equations by factoring relies fundamentally on this fact. Let us start with a review of how to factor a quadratic expression into two binomials. Here’s how it works. Let’s say that we have a quadratic equation of the form,
such that the left hand side can be written as the product of two simple binomials like this:
We call each of the expressions in parentheses binomials because they are each a polynomial with two terms. Factoring a quadratic expression in this manner is like “F.O.I.L.ing” in reverse (for a review of the F.O.I.L. method for multiplying binomials, refer to the tutorial entitled Multiplying Binomials). For such a factorization to work the following must hold:
If it seems like generating values for k, h, m and n which satisfy the above conditions could be very hard, you’re right. Factoring a quadratic can sometimes be a very difficult task and unfortunately there are few hard and fast rules to help with the process. Becoming skilled at factoring quadratics just takes practice.
Factor the quadratic expression:
One method for factoring a quadratic expression which many find helpful is as follows. The first step is to write down a set of two parentheses without even writing anything inside:
Note that the first term of the quadratic expression is or x times x, so a good choice for the first terms in each set of parentheses would be x. Like this:
Now note that because of the way that binomial multiplication (F.O.I.L.ing) works, the numbers we choose to add to each x must sum to positive 5 (the coefficient of x) and multiply to positive 6 (the constant term). With a little thought, one can see that the numbers positive 2 and positive 3 fit the bill; they sum to 5 and multiply to 6. So we conclude that the proper factorization is:
The usefulness of such a factorization becomes clear when we combine this process with the mathematical fact stated at the beginning. Notice that if we are successful in factoring a quadratic expression into the product of two binomials, then the original quadratic equation becomes:
Then, by the mathematical fact stated at the beginning we know that either
or both equal zero. Now these are just two linear equations and as such are very easy to solve (for a review of this topic, refer to the tutorial entitled Solving Linear Equations). The two solutions of the quadratic equation are then:
Often, the coefficients k and h are just 1 as in Example 1 and the solutions are even simpler.
Let’s return briefly to the first example. Suppose we took the example a little further and asked the question: “Solve the quadratic equation .” Once we generate the factorization, the problem is much simpler since we now need only to solve the equivalent problem:
Now applying the fact at the beginning, we can conclude that either,
So the two solutions are , and . Here’s another example which is a little tougher.
We start by factoring the left hand side into the product of two binomials keeping in mind the three requirements we have for the coefficients that were listed above. The factorization (which can be checked by F.O.I.L.ing) is this:
So the equivalent equation is
So the solutions are and .