Difference of squares or perfect-square trinomial?
Recognize the pattern from term count, signs, square roots, and the middle coefficient—then expand to confirm rather than trusting appearance alone.
A practical comparison that turns a vague choice into a repeatable test.
Updated July 13, 2026Two squared terms with subtraction suggest a difference of squares. Three terms suggest a perfect square only when the middle term is exactly ±2ab.
Difference of squares needs subtraction
Both terms must be perfect squares and the operation between them must be subtraction. A sum of squares does not factor into real linear factors by this pattern.
Identify the square roots, then write the conjugate pair with opposite signs.
Perfect-square trinomials need the middle check
The first and last terms may be squares without the trinomial being a perfect square. Multiply their roots, double the product, and compare with the middle term.
The sign of the middle term chooses the sign inside the repeated binomial.
Factor the GCF before pattern matching
A common factor can hide the pattern or make the apparent square roots misleading. Remove it first and inspect what remains.
Patterns can repeat: after a difference of squares, one factor may itself factor again.
Factor 4x² − 20x + 25.
Find the outer square roots.
The doubled product matches the middle term.
Use the negative sign from the middle term.
Expand to verify.
Common mistakes
- Factoring a sum of squares as real conjugates.
- Checking only the first and last terms of a trinomial.
- Stopping before removing a common factor.
Three takeaways
- Count terms and inspect signs.
- Verify the ±2ab middle coefficient.
- Factor completely, not just once.