How to Factor a Quadratic Expression: Step-by-Step Guide with Examples
To factor a quadratic expression (ax² + bx + c), first factor out any common factor, then find two binomials that multiply back to it. When a = 1, look for two numbers that multiply to c and add to b. When a is not 1, use grouping. For example, x² + 5x + 6 factors to (x + 2)(x + 3), because 2 and 3 multiply to 6 and add to 5.
Factoring a quadratic means rewriting it as a product of two simpler expressions, usually two binomials that multiply back to the original. It is one of the most useful skills in algebra, because it is how you solve quadratic equations, simplify fractions that contain variables, and find where a parabola crosses the x-axis.
The method you use depends on what the quadratic looks like: whether the terms share a common factor, whether the leading coefficient is 1, and whether it fits a special pattern like a difference of squares. This guide walks through each case with worked examples, shows how to pick the right method quickly, and covers how to check your answer.
What Does Factoring a Quadratic Mean?
A quadratic expression has the standard form ax² + bx + c, where a, b, and c are numbers and a is not 0. Factoring it means writing it as a product of factors, usually two binomials (px + q)(rx + s), whose product is the original expression.
The nice thing about factoring is that it comes with its own answer key: if you multiply your factors back out and get the original expression, your factoring is correct. Every worked example below ends with that check.
Step 1: Always Factor Out the GCF First
Before anything else, look for a greatest common factor, a number or variable that every term shares. If there is one, factor it out first, because it makes everything after it simpler.
Example: factor 4x² + 8x
Both terms share a factor of 4x. Pull it out:
4x² + 8x = 4x(x + 2)
Check: 4x times x is 4x², and 4x times 2 is 8x, so 4x(x + 2) = 4x² + 8x. Correct.
Sometimes the expression inside the parentheses can still be factored further, so after taking out the GCF, keep going if what is left is a factorable trinomial or a special case.
Factoring Trinomials When a = 1
For a trinomial in the form x² + bx + c, find two numbers that multiply to c and add to b. Those two numbers become the constants in your binomials.
Example: factor x² + 5x + 6
Find two numbers that multiply to 6 and add to 5: 2 and 3.
x² + 5x + 6 = (x + 2)(x + 3)
Check: (x + 2)(x + 3) = x² + 3x + 2x + 6 = x² + 5x + 6. Correct.
A quick note on signs. If c is positive, both numbers have the same sign as b. If c is negative, the two numbers have opposite signs.
Example with a negative: factor x² - x - 6
Here c is negative, so the numbers have opposite signs. Find two numbers that multiply to -6 and add to -1: -3 and 2.
x² - x - 6 = (x - 3)(x + 2)
Check: (x - 3)(x + 2) = x² + 2x - 3x - 6 = x² - x - 6. Correct.
Factoring Trinomials When a Is Not 1
When the leading coefficient is not 1, the shortcut above does not work directly. Use grouping, sometimes called the ac-method.
Example: factor 2x² + 7x + 3
- Multiply a and c: 2 times 3 = 6.
- Find two numbers that multiply to 6 and add to b, which is 7: the numbers 1 and 6.
- Split the middle term 7x into 1x + 6x: 2x² + x + 6x + 3.
- Group into pairs and factor each pair: x(2x + 1) + 3(2x + 1).
- Both groups share (2x + 1), so factor it out: (2x + 1)(x + 3).
Check: (2x + 1)(x + 3) = 2x² + 6x + x + 3 = 2x² + 7x + 3. Correct.
Special Case: Difference of Squares
If you see two perfect squares separated by a minus sign, it factors instantly. The pattern is:
a² - b² = (a - b)(a + b)
Example: factor x² - 9
Here x² and 9 are both perfect squares (9 is 3²), so:
x² - 9 = (x - 3)(x + 3)
The same works with coefficients. For example, 4x² - 25 = (2x - 5)(2x + 5), since 4x² is (2x)² and 25 is 5². One warning: a sum of squares, a² + b², does not factor over the real numbers.
Special Case: Perfect Square Trinomials
Some trinomials are the square of a single binomial. The patterns are:
a² + 2ab + b² = (a + b)² a² - 2ab + b² = (a - b)²
You can spot one when the first and last terms are perfect squares and the middle term is twice the product of their roots.
Example: factor x² + 6x + 9
Both x² and 9 are perfect squares, and the middle term 6x is twice 3 times x, so:
x² + 6x + 9 = (x + 3)²
Check: (x + 3)² = (x + 3)(x + 3) = x² + 6x + 9. Correct. The same pattern with a minus gives a difference form, for example x² - 10x + 25 = (x - 5)².
Which Method Should You Use?
Work through the quadratic in this order and the right method becomes obvious:
- Factor out the GCF first, always.
- Count the terms. Two terms usually means a difference of squares. Three terms means a trinomial.
- For a trinomial, check for a perfect square first, since it is the fastest to spot. If it is not one, use the multiply-to-c-add-to-b shortcut when a = 1, and grouping when a is not 1.
- Check by multiplying back every time.
How to Check Your Factoring
Multiply your factors back out, using FOIL for two binomials, and confirm you get the original expression. If you do, the factoring is correct. If you do not, you have a sign or number error to track down, which is almost always in the two numbers you chose. This check takes a few seconds and catches nearly every mistake.
Common Mistakes to Avoid
- Skipping the GCF. Always pull out the greatest common factor before trying anything else.
- Sign errors. Choosing the wrong signs for your two numbers is the most common error, especially when c is negative.
- Treating a sum of squares like a difference. a² + b² does not factor over the real numbers, only a² - b² does.
- Stopping too early. After the GCF, the expression inside the parentheses may still factor further.
- Not checking. Multiplying the factors back out is the fastest way to confirm your answer.
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Frequently Asked Questions
What does it mean to factor a quadratic expression? It means rewriting a quadratic in the form ax² + bx + c as a product of two simpler expressions, usually two binomials, that multiply back to give the original. For example, x² + 5x + 6 factors to (x + 2)(x + 3).
How do you factor a trinomial when a = 1? Find two numbers that multiply to c and add to b, then use them as the constants in your binomials. For x² + 5x + 6, the numbers 2 and 3 multiply to 6 and add to 5, giving (x + 2)(x + 3).
How do you factor a quadratic when a is not 1? Use grouping. Multiply a and c, find two numbers that multiply to that product and add to b, split the middle term, and factor by grouping. For 2x² + 7x + 3, this gives (2x + 1)(x + 3).
What is the difference of squares? It is the pattern a² - b² = (a - b)(a + b). For example, x² - 9 factors to (x - 3)(x + 3). A sum of squares, a² + b², does not factor over the real numbers.
Why is factoring quadratics useful? Factoring is how you solve quadratic equations using the zero product property, simplify rational expressions, and find where a parabola crosses the x-axis, so it is a foundation for a lot of later algebra.