x² + 6x + 9 looks like an ordinary trinomial. It is not. It is (x + 3)² in disguise, and once you learn to spot the disguise, problems like this take five seconds.

The two checks

Take x² + bx + c. It is a perfect square trinomial only if both of these hold. First, the first and last terms must be perfect squares. For x² + 6x + 9 that means √x² = x and √9 = 3. Second, the middle term must equal twice their product: 2 × x × 3 = 6x, which matches.

Both checks pass, so x² + 6x + 9 = (x + 3)².

The sign of the middle term tells you the sign inside the brackets. A plus in the middle gives (x + 3)², a minus would give (x − 3)². In general: a² + 2ab + b² = (a + b)² and a² − 2ab + b² = (a − b)².

Example: 4x² − 12x + 9

First check: √(4x²) = 2x and √9 = 3, both clean. Second check: 2 × 2x × 3 = 12x, which matches the middle term. The middle term is negative, so the answer is (2x − 3)². Expand to verify: (2x − 3)(2x − 3) = 4x² − 6x − 6x + 9 = 4x² − 12x + 9. Correct.

False alarms

Not every trinomial with square ends is a perfect square. The middle term decides.

x² + 5x + 9 has square ends, but 2 × x × 3 = 6x, not 5x. It fails the second check, so factor it the normal way on the simple trinomial calculator.

9x² + 12x + 7 fails immediately because √7 is not clean.

x² + 6x + 8 is not a square either, since 8 is not a perfect square. It factors as (x + 2)(x + 4), but that is regular trinomial factoring.

The classic mistake is seeing square first and last terms and skipping the middle-term check. That second check is the one that matters most.

When the leading coefficient is not 1

Same two checks, no new rules. Take 25x² + 40x + 16: √(25x²) = 5x, √16 = 4, and 2 × 5x × 4 = 40x. Both pass, so it is (5x + 4)². For trinomials that are not squares, the ac-method walkthrough in our step-by-step factoring calculator covers those.

Related patterns

A difference of squares (a² − b²) is the mirror image of this pattern, covered in the difference of squares guide. Four-term polynomials go to the factoring by grouping calculator. All tools live on the calculators page.