Solve Quadratic Functions By Factoring
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.
