SAT Math: How to Factor Quadratic Equations

On the SAT, you may be asked to solve quadratic equations or be required to factor an equation in order to make a comparison. Before you bust out the quadratic formula, relax: no questions on the SAT require you to know or use the quadratic formula. Instead, if you need to find a value in a quadratic, you should be able to factor the equation in order to find it.

Factoring quadratic equations is much easier than it sounds. Below we breakdown quadratic equations and how to factor them.

 

Quadratic equations

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

3t2t = 0

 

Factoring quadratic equations 

First put the equation in the ax2 + bx + c form. Once the equation is in this form, turn your attention to the a, b, and c terms. Usually a will equal 1. If it doesn’t, no sweat; we’ll explain what to do later, but for now let’s say a = 1.

Next take a look at b and c and try to come up with two numbers whose product is c and whose sum is b. This last step is the key to factoring. Let’s walk through an example.

In the first equation from above, 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 be either -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

and

y = -2 or -3

 

There you have it! Now you know quadratic equations are nothing to be intimidated by.