Solve Quadratic Functions By Factoring
Quadratic equations by factoring — broken down step by step until it makes sense.
Factoring quadratic equations is one of those topics that feels impossible until something clicks — and then it all falls into place. These guided notes and practice sets are designed to get you to that moment faster, walking you through the method one clear step at a time before giving you plenty of problems to practice on your own.
What's inside:
Two versions of guided notes: a fill-in-the-blank version to use while working through the material actively, and a completed version to use as a reference when reviewing or checking your understanding
3 progressive practice sets that build in difficulty:
Factoring when the leading coefficient is 1 (the standard starting point)
Factoring when the leading coefficient isn't 1 (where most students get stuck)
Special patterns, including difference of squares and perfect square trinomials
Full answer keys for every problem so you can check your work and understand where you went wrong without needing anyone else to mark it for you
Perfect for:
Students in Grades 9–12 taking Algebra 1 or working through quadratic equations for the first time
Catching up after a confusing lesson or filling gaps before a test
Anyone who needs more structured practice than a textbook provides
Parents supporting their child through a tricky algebra topic at home
The two versions of notes make this flexible. Use the fill-in-the-blank set when you're learning it fresh, then keep the completed notes as a reference when you're doing homework or revising later. Everything you need is self-contained, so you can work through it entirely independently.
In our last post, we learned how to factor quadratic functions. While not every quadratic in standard form can be factored, when it can, factoring is a powerful method for solving.
When solving a quadratic function, we are looking for the values of x when f(x) = 0. These values are called the x-intercepts, and you may also hear them referred to as the zeroes or the roots of the function.
A quadratic can have:
Two real solutions → the graph crosses the x-axis twice
One real solution → the graph touches the x-axis once (a repeated root)
No real solutions → the graph does not touch the x-axis at all
Factoring helps us find these solutions when possible.
If you would like a refresher on how to factor quadratic functions, check out our post here!
Steps to Solve by Factoring
1. Set the quadratic equal to zero
Because we’re solving for the x-intercepts, we must start by setting the quadratic equal to zero.
Example 1:
f(x) = x2 + 5x + 6
0 = x2 + 5x + 6
2. Factor the quadratic
Use the factoring strategies we practiced in the previous post.
0 = (x+2)(x+3)
3. Apply the Zero Product Property
The Zero Product Property states:
If a • b=0, then a = 0 or b = 0
Since our quadratic is written as a product of two factors, we can set each factor equal to zero:
0 = (x + 2)(x + 3)
0 = x + 2 and 0 = x + 3
x = −2 and x = −3
So, the x-intercepts are (−2,0) and (−3,0).
More Examples
Example 2:
y = 2x2 + 8x
0 = 2x2 + 8x
0 = 2x(x + 4)
0 = 2x and 0 = x + 4
x = 0 and x = −4
The x-intercepts are (0,0) and (−4,0).
Example 3:
y = 2x2 + 7x + 3
0 = 2x2 + 7x + 3
0 = (2x + 1)(x + 3)
0 = 2x + 1 and 0 = x + 3
x = -1/2 and x = −3
The x-intercepts are (−1/2,0) and (−3,0).
Example 4:
y = x2 + 4x + 4
0 = x2 + 4x + 4
0 = (x + 2)(x + 2)
0 = x + 2 and 0 = x + 2
0 = x + 2
x = -2
The x-intercept is (-2, 0).
Notice that in this case there was only one solution because the quadratic is a perfect square trinomial.
Common Mistakes to Watch For
Not setting the equation equal to zero first
Example: Trying to factor y = x^2 + 5x + 6 without writing 0 = x^2 + 5x + 6.
Always remember: we’re solving for where y = 0.
Forgetting to solve both factors
Some students stop after finding just one solution.
If you have (x + 2)(x + 3)=0, you must solve both: x + 2 = 0 and x + 3 = 0.
Dropping a repeated solution
In cases like (x + 2)(x + 2)=0, both factors give the same root, x = −2.
That doesn’t mean there are two different solutions—just one root with multiplicity two.
Sign mistakes
Example: Solving x + 3 = 0 but writing x = 3 instead of x = −3.
Double-check when moving terms across the equals sign.
