Algebra · Step-by-step guide
Factoring Trinomials
Factor quadratic trinomials when the leading coefficient is 1 or another integer, with reliable checking steps.
Start with the central idea
To factor , find two numbers whose product is c and sum is b. For , use the ac method: split the middle term using numbers with product ac and sum b, then factor by grouping.
How it works
Build the method from meaning before memorizing the moves.
These are the decisions that make the procedure reliable. Read them once, then look for each idea in the worked examples below.
Take out the GCF first
A greatest common factor belongs outside the parentheses and makes the remaining trinomial simpler.
Use product and sum
The factor constants multiply to the final term while their cross-products combine to the middle term.
Check by multiplying
FOIL or distribute the factors. The original three terms should reappear exactly.
See the structure
A picture makes the relationships easier to remember.
Worked examples
Follow the reason for each line, then try to reproduce it without looking.
Factor .
- Find numbers with product 12 and sum 7.
- The numbers are 3 and 4.
- Write the corresponding binomials.
Factor .
- The product is −12 and sum is −1.
- Use −4 and 3.
- Check the middle terms: .
Factor .
- ; use 9 and 2.
- Rewrite as .
- Group: .
Common mistakes—and how to avoid them
Accuracy improves fastest when you know what to check.
Skipping a common factor.
Using numbers with the right product but the wrong sum.
Failing to verify the result by multiplication.
Turn the explanation into a skill
Reading creates recognition. Independent practice creates recall.
Factoring Trinomials
Practice monic trinomials, non-monic trinomials, and sign patterns.
Preview the worksheetA strong practice loop: solve one example with the guide open, solve a similar problem without it, explain the method aloud, and return the next day for a short mixed review.
Practice problems with worked solutions
Solve each problem on paper, use the hint only if needed, then compare every step.
Factor .
Show hint
Find factors of 20 that add to 9.
Show worked solution
- The numbers are 4 and 5.
- Write one binomial for each.
- Check: .
Answer:
Factor .
Show hint
Find numbers with product −15 and sum −2.
Show worked solution
- Use −5 and 3.
- Write .
- The cross terms combine to .
Answer:
Factor .
Show hint
Use the ac method.
Show worked solution
- ; use 6 and 1.
- .
- Group to get .
Answer:
Factor .
Show hint
Take out the GCF.
Show worked solution
- The GCF is .
- Divide both terms by .
- .
Answer:
Factor .
Show hint
Use numbers with product −24 and sum −10.
Show worked solution
- Use −12 and 2.
- .
- Group: .
Answer:
Frequently asked questions
Quick answers before you move on.
What should I remember first?
To factor , find two numbers whose product is c and sum is b. For , use the ac method: split the middle term using numbers with product ac and sum b, then factor by grouping.
How do I know whether I understand this topic?
You should be able to name the method, explain why each step is valid, complete a new example without copying, and check whether your answer is reasonable.
What should I do if I keep making the same mistake?
Write the error as a specific checkpoint—for example, “I will identify the hypotenuse before substituting.” Then use that checkpoint on three short problems rather than repeating a full page without feedback.