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!

MATH SEATWORK

Seatwork Time!

Answer all 5 questions. Good luck!

Please select an answer first!

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Made for students who want to master Factoring Perfect Square Trinomials