Always Start by Extracting the GCF
Before attempting any other factoring technique, always check whether every term in the quadratic shares a greatest common factor (GCF). If they do, factor it out first.
Consider 6x² + 18x + 12. Every coefficient is divisible by 6, so pull it out:
6(x² + 3x + 2)Now you have a much simpler trinomial to factor inside the parentheses.
This step is critical because skipping it makes the remaining factoring significantly harder — or in some cases, impossible to see.
For example, 4x² + 8x − 12 factors as 4(x² + 2x − 3) = 4(x + 3)(x − 1).
Without extracting the 4 first, you would need to wrestle with larger numbers.
The GCF can also include a variable:
3x³ + 9x² − 12x = 3x(x² + 3x − 4) = 3x(x + 4)(x − 1)Make extracting the GCF your automatic first step for every factoring problem. This single habit will simplify your work dramatically.
Factoring Simple Trinomials (a = 1)
x² + bx + c — where the leading coefficient is 1 — factoring is straightforward. You need to find two numbers that multiply to c and add to b.For example, factor x² + 7x + 12. You need two numbers that multiply to 12 and add to 7. Those numbers are 3 and 4, so the factored form is:
(x + 3)(x + 4)When c is negative, one of the two numbers will be negative. Factor x² + 2x − 15: you need two numbers that multiply to −15 and add to 2. Those are 5 and −3, giving:
(x + 5)(x − 3)When both b and c are negative, like x² − 5x − 6, you need numbers that multiply to −6 and add to −5: those are −6 and 1, giving:
(x − 6)(x + 1)Practice this method with many examples until identifying the two numbers becomes almost automatic. Solver AI can verify your factoring instantly — just type in the expression to check your work.
The AC Method for Trinomials (a > 1)
When the leading coefficient is not 1, the simple method above does not work directly. Instead, use the AC method (also called factoring by grouping).
- Multiply a × c
For
ax² + bx + c, multiply a times c to get the product AC. - Find two numbers
Then find two numbers that multiply to AC and add to b.
- Rewrite and group
Rewrite the middle term using these two numbers, and factor by grouping.
2x² + 7x + 3Let's factor 2x² + 7x + 3. Here, AC = 2 × 3 = 6. We need two numbers that multiply to 6 and add to 7: those are 6 and 1.
Rewrite: 2x² + 6x + 1x + 3.
Group: (2x² + 6x) + (x + 3) = 2x(x + 3) + 1(x + 3) = (2x + 1)(x + 3).
(2x + 1)(x + 3)3x² − 10x + 8Another example: 3x² − 10x + 8. AC = 24, and we need two numbers that multiply to 24 and add to −10: those are −6 and −4.
Rewrite: 3x² − 6x − 4x + 8 = 3x(x − 2) − 4(x − 2) = (3x − 4)(x − 2).
(3x − 4)(x − 2)The AC method works for every factorable trinomial, making it the most versatile technique in your toolkit.
Difference of Squares
a² − b², which factors as (a + b)(a − b). Note the key requirements: it must be a subtraction (not addition) of two perfect squares.For example, x² − 25 = (x + 5)(x − 5) because x² is the square of x and 25 is the square of 5.
Similarly, 4x² − 9 = (2x + 3)(2x − 3) and 16x² − 49y² = (4x + 7y)(4x − 7y).
A common pitfall: x² + 25 is a sum of squares, which cannot be factored over the real numbers. Students sometimes incorrectly write (x + 5)(x − 5) for this — but that produces x² − 25, not x² + 25.
Also watch for hidden differences of squares:
x⁴ − 1 = (x² + 1)(x² − 1) = (x² + 1)(x + 1)(x − 1)Always check if any factor can be factored further.
Recognizing this pattern instantly will save you valuable time on exams.
Perfect Square Trinomials
a² + 2ab + b² = (a + b)² and a² − 2ab + b² = (a − b)². The key signature is that the first and last terms are perfect squares, and the middle term is exactly twice the product of their square roots.x² + 10x + 25For instance, x² + 10x + 25: the first term x² is a perfect square, the last term 25 = 5² is a perfect square, and the middle term 10x = 2(x)(5). So this factors as:
(x + 5)²9x² − 12x + 4Another example: 9x² − 12x + 4. Here 9x² = (3x)², 4 = 2², and 12x = 2(3x)(2). So it factors as:
(3x − 2)²Recognizing perfect square trinomials is especially useful when completing the square to solve quadratic equations or when simplifying expressions in calculus. To verify, always expand your answer: (3x − 2)² = 9x² − 12x + 4.
If you are ever unsure whether a trinomial is a perfect square, use Solver AI to check — it will show you the factored form and confirm the pattern.
A Decision Tree for Factoring
With five factoring techniques available, knowing which to use can feel overwhelming. Follow this decision tree for any quadratic expression.
- Step 1 — Extract the GCF
Extract the GCF from all terms. Always do this first.
- Step 2 — Count the terms (binomial)
Count the terms. If you have a binomial (two terms), check for the difference of squares pattern (a² − b²).
- Step 3 — Check for perfect square trinomial
If you have a trinomial (three terms), check whether the first and last terms are perfect squares with the middle term equal to 2ab — if so, use the perfect square trinomial pattern.
- Step 4 — Simple trinomial (a = 1)
If it is a trinomial with leading coefficient 1, find two numbers that multiply to c and add to b.
- Step 5 — AC method (a > 1)
If the leading coefficient is greater than 1, use the AC method (factor by grouping).
- Step 6 — Factor again or call it prime
After factoring, check whether any resulting factor can be factored further. If no technique applies, the expression may be prime (unfactorable over the integers).
Always verify your answer by expanding the factors back out — the product should equal the original expression.
With practice, this decision process becomes instinctive, and you will factor most quadratics in under a minute.