Factoring Trinomials Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Factoring Trinomials.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Factoring a trinomial of the form ax^2 + bx + c into a product of two binomials by finding two numbers that multiply to ac and add to b.

You are reverse-engineering FOIL. If (x + p)(x + q) = x^2 + (p+q)x + pq, then you need two numbers p and q whose sum is b and whose product is c (when a = 1). When a \neq 1, use the AC method: find two numbers that multiply to ac and add to b, then split the middle term and factor by grouping.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Finding the right pair of numbers is the key—they must simultaneously satisfy both a sum and a product condition.

Common stuck point: When a \neq 1, the simple 'find two numbers' method must be extended to the AC method or trial and error with the leading coefficient.

Sense of Study hint: Write two columns: factor pairs of the constant (or ac), and their sums. Find the pair whose sum equals b.

Worked Examples

Example 1

easy
Factor x^2 + 5x + 6.

Solution

  1. 1
    Step 1: Find two numbers that multiply to 6 and add to 5.
  2. 2
    Step 2: 2 \times 3 = 6 and 2 + 3 = 5. The numbers are 2 and 3.
  3. 3
    Step 3: Factor: (x + 2)(x + 3).
  4. 4
    Check: (x+2)(x+3) = x^2 + 3x + 2x + 6 = x^2 + 5x + 6 ✓

Answer

(x + 2)(x + 3)
For x^2 + bx + c, find two numbers p, q where pq = c and p + q = b. Then factor as (x+p)(x+q). This reverses the FOIL multiplication process.

Example 2

hard
Factor 2x^2 + 7x + 3 using the AC method.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Factor x^2 - 7x + 12.

Example 2

medium
Factor x^2 + 2x - 15.
← Back to Factoring Trinomials Formula Explained Practice Mode

Background Knowledge

These ideas may be useful before you work through the harder examples.

factoringpolynomial multiplication