Geometry, algebra, number theory – studying for the GMAT can reintroduce to you to subjects you haven’t come across since high school. And though you’ve learned them once, and were maybe even good at them, most test takers could use a refresher to help brush off the mental dust that’s accumulated over the years.
Factoring quadratic equations is one such skill you were likely once proficient at but which now may leave you scratching your head. Factoring is one method by which to solve quadratic equations, and if you hope to solve some of the GMAT’s tougher problems, you’re going to need to know how to do it.
A quadratic equation is a specific type of equation that contains one variable raised to the second power (e.g. x2) and one variable raised to the first power (e.g. x). Here are some examples of quadratic equations:
x2 + 2x – 8 = 0
5y2 + 15y = -30
t = 3t2
You may remember the famous quadratic equation form:
ax2 + bx + c
where the a, b, and c are integers. This is just a particular form you can arrange any quadratic equation into. For example, the first equation above is already in this form, but you could rearrange the following two equations as follows:
5y2 + 25y + 30 = 0
3t2 – t = 0
Factoring quadratic equations
Remembering the ax2 + bx + c form is important because putting a quadratic formula into this form is the first step in factoring it. Once the equation is in this form, turn your attention to the a, b, and c terms. Usually a will equal 1. If so, look at b and c and try to come up with two numbers whose product is c and whose sum is b. This is the key step in factoring.
In the first example, x2 + 2x – 8 = 0, the numbers 4 and -2 have a product of -8 (c) and a sum of 2 (b).
Once you’ve found the two numbers, put them in the form:
(x + number)(x + number)
In our example, that would be:
(x + 4)(x – 2) = 0
To solve the equation, we know that (x + 4) must equal 0 or (x – 2) must equal 0. Therefore, x could either be -4 or 2. These are the roots of the equation, the numbers that make the equation true.
When a doesn’t equal 1
If a does not equal one, you’ll need to divide the equation by a. For example, let’s look at the second quadratic equation: 5y2 – 15y = 30
After you’ve put it into ax2 + bx + c form, divide by the a term.
5y2 + 25y + 30 = 0
Divide through by 5 to get:
y2 + 5y + 6 = 0
What are two numbers whose sum is 5 and product is 6? How about 3 and 2? Those work. So continuing:
(y + 2)(y + 3) = 0
y = -2 or -3
GMAT questions will never ask you directly to factor quadratic questions. Instead, factoring will be a necessary step to arrive at the correct answer. Watch out for quadratics anytime you’re dealing with variables in an equation, no matter if it’s a geometry problem, a word problem, or a seemingly straightforward algebra problem.