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FACTORING: PERFECT SQUARE TRINOMIAL

MATH LESSON

FACTORING: PERFECT SQUARE TRINOMIAL

A trinomial that is the result of squaring a binomial. Recognize the pattern and factor it in seconds!

A Perfect Square Trinomial has exactly 3 terms where the first and last are perfect squares, and the middle term is exactly twice the product of their square roots.
a² + 2ab + b² = (a + b)²  a² − 2ab + b² = (a − b)²
a² + 2ab + b²
=
(a + b)²
a² − 2ab + b²
=
(a − b)²
How to Recognize a Perfect Square Trinomial

Check ALL three conditions before factoring:

1. It must be a trinomial (exactly 3 terms)
2. First and last terms must be perfect squares
3. Middle term = 2 × √(first) × √(last)
The sign of the middle term tells you which pattern to use: + → (a+b)²  |  − → (a−b)²
The "Double and Multiply" Check

This is the most important step. After finding √(first) and √(last), verify the middle term:

Test: x² + 6x + 9
√(x²) = x  |  √(9) = 3
Check: 2 × x × 3 = 6x ✓ MATCHES!
Test: x² + 5x + 9
√(x²) = x  |  √(9) = 3
Check: 2 × x × 3 = 6x  ≠ 5x  × NOT a PST!
Two Patterns — Positive vs. Negative

The sign of the middle term determines the sign inside the binomial factor:

a² + 2ab + b² = (a + b)²
Positive middle term → BOTH signs are positive
a² − 2ab + b² = (a − b)²
Negative middle term → BOTH signs are negative
Last term is always POSITIVE in a PST!
Common Mistakes!

Mistake 1: Forgetting to check the middle term (the 2ab part).

x² + 5x + 9 = (x + 3)² × WRONG
2 × 3 × 1 = 6 ≠ 5, so NOT a PST! ✓

Mistake 2: Mixing up signs in the factor.

x² − 6x + 9 = (x + 3)(x − 3) × WRONG (that's DOTS!)
x² − 6x + 9 = (x − 3)² ✓ SAME sign: (x − 3)(x − 3)

Mistake 3: Forgetting to square root the coefficient.

4x² + 12x + 9 = (4x + 3)² × WRONG
4x² + 12x + 9 = (2x + 3)² ✓ √4 = 2, √9 = 3
Worked Examples
Example 1: Factor: x² + 6x + 9
√(x²) = x  |  √(9) = 3
Check: 2 × x × 3 = 6x ✓
→ (x + 3)²
Example 2: Factor: x² − 10x + 25
√(x²) = x  |  √(25) = 5
Check: 2 × x × 5 = 10x ✓ (negative middle → minus)
→ (x − 5)²
Example 3: Factor: 4x² + 12x + 9
√(4x²) = 2x  |  √(9) = 3
Check: 2 × 2x × 3 = 12x ✓
→ (2x + 3)²
Example 4: Factor: 16x² − 24x + 9
√(16x²) = 4x  |  √(9) = 3
Check: 2 × 4x × 3 = 24x ✓ (negative middle → minus)
→ (4x − 3)²
Example 5: Factor: 25x² + 20x + 4
√(25x²) = 5x  |  √(4) = 2
Check: 2 × 5x × 2 = 20x ✓
→ (5x + 2)²

Shortcut: √(first) → a, √(last) → b. Check if 2ab = middle term. If yes, write (a + b)² or (a − b)² based on the sign!

LEARNING CHECK

Let's see what you learned!

Answer these 2 questions correctly to proceed to the Quiz. If wrong, questions will be reshuffled and you try again!
Perfect! Both answers are correct! Proceeding to the Quiz...
Not quite! The questions will be reshuffled. Read the lesson again and try!
Answer all questions first!
MATH QUIZ

Quiz Time!

10 questions about Factoring Perfect Square Trinomials. You got this!

Answer all questions first before submitting!
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Made for students who want to master Factoring Perfect Square Trinomials

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