Factoring Intuition Examples in Math

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

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

Concept Recap

Understanding factoring as finding what multiplies together to give an expression.

Reverse engineering multiplication: 'What times what gives x^2 + 5x + 6?'

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: Factoring is 'un-distributing'—reversing the multiplication process.

Common stuck point: Not all quadratic expressions factor over integers — when no integer pair works, use the quadratic formula instead.

Sense of Study hint: List all pairs of integers that multiply to give the constant, then check which pair adds to the middle coefficient.

Worked Examples

Example 1

easy
What two numbers multiply to 12 and add to 7?

Solution

  1. 1
    List factor pairs of 12: (1,12), (2,6), (3,4).
  2. 2
    Check sums: 1+12=13, 2+6=8, 3+4=7 ✓
  3. 3
    The numbers are 3 and 4.

Answer

3 \text{ and } 4
Factoring intuition is about finding what multiplies together to give an expression. For trinomials x^2 + bx + c, you need two numbers with product c and sum b.

Example 2

medium
Find two numbers that multiply to -15 and add to 2.

Practice Problems

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

Example 1

easy
Find two numbers that multiply to 18 and add to 9.

Example 2

hard
Find two numbers that multiply to -24 and add to -2.
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Background Knowledge

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

multiplicationdistributive property