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How to Factor Every Type of Quadratic Expressions (with Examples)

Updated: Sep 12, 2023

Factoring can be challenging with different numbers of terms and special cases, but this post demonstrates how to factor all quadratic expressions. Factoring is the most important skill for Algebra students to master as it comes back in every course of high school (and even college!) math.

Three years ago, I started tutoring a student at the end of her 9th grade Algebra 1 class. She was struggling to learn how to factor quadratic expressions. She couldn’t identify where to start as each problem looked slightly different and there were different “rules” to follow. We broke down these factoring rules based on the number of terms, outlined below.

I told her factoring was a key skill she would use in upcoming years of Algebra 2, Geometry, Pre-Calculus and beyond so I wanted to make sure she felt confident in factoring all of the forms.

What is factoring a quadratic expression?

Factoring is rewriting a quadratic expression into the product of two linear factors.

How do you factor quadratic expressions?

This depends on the number of terms of the expression. I’ll explain the steps for factoring quadratics of various terms and follow each with examples.

For Binomials with 2 terms, we first look for a Greatest Common Factor (GCF), a multiple that both terms have in common. If that exists, factor it out. If not, it may be a special case called Difference of Squares. Difference means subtraction so it’s two squared terms being subtracted. Rewrite the expression as a^2-b^2 then it factors to (a-b)(a+b) as shown in the example on the right below.

Factoring Quadratic Binomial Expressions

Notice below the problem on the right (in blue writing), after I used difference of squares to factor, you can also use the box method to solve. It’s a longer process but if you want to use the box method, you don’t have to memorize the difference of squares shortcut. The box method process is used for factoring trinomials.

For Trinomials with 3 terms, I also first look for a GCF to factor out, if possible. Then, with the quadratic in standard ax^2+bx+c form, we need a pair of numbers that multiply to the result of a*c and also add to b. We replace the middle bx term with the pair of numbers we just found. Applying that to the problem on the left below, we replace 5y with 12y and -7y. Finally, we use that result in either the area box method or factoring by grouping.

When I learned how to factor 20 years ago, I learned a method called factoring by grouping. Many teachers still use this method. The area box method is essentially the same process but it’s neater organization for some students. See the similarities in differences in the example below.

Factoring Quadratic Trinomial Expressions

One advantage of using factoring by grouping is it has a built-in check. Notice that once the GCF is factored out of both groups, the resulting factor, y+12, matches.

Also, there is a shortcut for both methods if a=1. Notice the pair of numbers that multiply to the result of a*c and add to b are the numbers in each factor. You can skip the box or grouping only if the leading coefficient is 1. This wouldn’t work for the example on the right, where the leading coefficient, a, is 6.

If you’re in Algebra 2 and factoring polynomials of higher degree (quadratics have a degree/power of 2), you have to factor a 4-term expression. This polynomial below is cubic since the highest power/degree is 3. If you’re only factoring quadratic expressions, you may not need to worry about this case! The process is still to first check for a GCF but then you can jump right into factoring by grouping or the box method.

Factoring Polynomial 4 terms

Struggles of factoring:

Notice in the example above, we can factor further if a quadratic expression remains. In other words, if there is still an x^2, it can possibly factor further. This can get tricky when multiple methods are involved.

Another challenge for students is identifying common factors in order to factor out the Greatest Common Factor. To help with this, create a factor tree for each term. Below, I circled the common factors to determine the GCF.

Factoring Quadratics Greatest Common Factor Tree

Why do we need factoring?

There are two main times factoring is used with quadratic functions:

1. Solving quadratic equations.

2. Finding x-intercepts on graph of a parabola.

These are technically the same thing. Solving x^2-3x+2=0 gives the x-intercepts for y= x^2-3x+2. When solving quadratic equations, factoring is just one method. Learn about the other methods for solving quadratic equations and when to use each method.

Factoring comes back in Algebra 2 with solving polynomial equations of higher powers and in Pre-Calculus with solving trigonometric equations.

Just recently, now that student I tutor is a senior, she randomly said to me, “You were right. Factoring really is the one concept that keeps coming up year after year!” Learn it now so you’ll be prepared for future math classes. If you need help mastering factoring, fill out our New Student Form and we’ll personalize individual tutoring sessions to ensure you’re confident in factoring all the various forms.

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