How to Factor Quadratic Functions
Quadratic functions are a type of polynomial function where the highest exponent is 2. They can be written in several different forms, and each form highlights different information.
In this post, we’ll focus on quadratics written in standard form:
f(x) = ax2 + bx + c, where a ≠ 0
Try it yourself: Graphing Quadratics Tool
Practice what you just read. Work through a parabola one decision at a time: find the vertex, decide which way it opens, build a table of points, and watch the curve appear on the coordinate plane as you go.
Open the Graphing ToolWhen quadratics are written in this way, we can find their x-intercepts (also called roots or zeros) using different solving methods:
Factoring
Quadratic Formula
Completing the Square
Square Root Method
We’ll cover all of these in future lessons, but today we’ll focus on factoring.
Remember: Solving a quadratic means finding the x-values where the graph crosses the x-axis. A quadratic can have 0, 1, or 2 x-intercepts.
Case 1: Factoring When There Is No “c” Term
If the quadratic looks like:
f(x) = ax2 + bx, where a ≠ 0
then the constant term c = 0. In this case, we can factor by taking out the Greatest Common Factor (GCF).
Factoring Out the GCF
The GCF is the largest factor shared by all terms—both the numbers (coefficients) and the variables.
Example:
f(x) = 2x2 + 8x
Coefficient GCF: between 2 and 8 → 2
Variable GCF: between x2 and x → x
So the overall GCF is 2x.
Now factor it out:
f(x) = 2x2 + 8x
f(x) = 2x(x + 4)
You can always check by distributing back to the original form.
An infographic titled "Factoring out the GCF" works through the example 2x squared plus 8x in three steps.
Case 2: Factoring When a = 1
If the quadratic looks like:
f(x) = x2 + bx + c
we can use the X-Method.
The X-Method
Identify the a, b, and c values.
Multiply a • c. Since a = 1, the product is just c. Write this value in the top of the X.
Write b in the bottom of the X.
Find two numbers that multiply to the top number and add to the bottom number.
Rewrite in factored form.
Example:
y = x2 + 5x + 6
a = 1, b = 5, and c = 6
Top of the X: 1 • 6 = 6
Bottom of the X: b = 5
Factor pair: 2 and 3 (since 2 • 3 = 6 and 2 + 3 = 5)
So,
y = (x + 2)(x + 3)
An infographic titled "The X-Method" works through the example x squared plus 5x plus 6 in five steps.
Case 3: Factoring When a ≠ 1
If a is not equal to 1, factoring takes an extra step. Two common approaches are:
The Box Method
The Grouping Method
Both start with the same first step: multiply a • c.
The Box Method
Example:
y = 2x2 + 7x + 3
a = 2, b = 7, and c = 3
Multiply: a • c = 2 ⋅ 3 = 6
Find the two numbers that multiply to 6 and add to 7 → 6 and 1
Now set up a 2×2 multiplication box:
| 2x | +1 | |
| x | 2x2 | 1x |
| +3 | 6x | 3 |
The factors that make this box work are:
(x + 3) and (2x + 1)
Our final factored form will be:
y = (x + 3)(2x + 1)
An infographic titled "The Box Method" works through the example 2x squared plus 7x plus 3 in seven steps.
The Grouping Method
Using the same example:
y = 2x2 + 7x + 3
Multiply a • c = 6, and find numbers that multiply to 6 and add to 7 → 6 and 1.
Rewrite the middle term (7x) as 6x +1x. The coefficients are the values found in Step 1.
y = 2x2 + 6x + 1x + 3
3. Group and factor each part using the GCFs:
y = (2x2 + 6x) + (1x + 3)
y = 2x(x+3) + 1(x+3)
4. Factor out the common binomial:
(2x + 1)(x + 3)
An infographic titled "Factor By Grouping" works through the example f of x equals 2x squared plus 7x plus 3 in seven steps.
Conclusion
Factoring quadratics is one of the most useful methods for finding x-intercepts. Which method you use depends on the structure of the quadratic:
Factoring out the GCF (when c = 0)
The X-Method (when a = 1)
The Box Method (when a ≠ 1)
Factor by Grouping (when a ≠ 1)
In our next posts, we’ll explore special factoring patterns like:
Perfect Square Trinomials
Difference of Squares
These patterns make factoring even faster!
