MATH LESSON

FACTORING: QUADRATIC TRINOMIAL

When it is not a perfect square, find two numbers that multiply to the last term and add to the middle term!

To factor x² + bx + c, find two numbers that multiply to c and add to b. Then write them as (x + p)(x + q).

x² + bx + c = (x + p)(x + q) where p × q = c and p + q = b

x² + bx + c

=

(x + p)

×

(x + q)

What is a Quadratic Trinomial?

It has 3 terms, the highest degree is 2, and the leading coefficient is 1 (x²). It is NOT a perfect square trinomial.

x² + 5x + 6 — YES, a quadratic trinomial!

x² + 6x + 9 — NO, this is a Perfect Square Trinomial!

x² − 16 — NO, this is a Difference of Two Squares!

The "Factor c, Add to b" Method

Look at x² + 5x + 6. Ignore the x² for now. We need two numbers that:

Multiply to c = 6

Add to b = 5

Factors of 6: (1, 6) → 1+6 = 7 ≠ 5
Factors of 6: (2, 3) → 2+3 = 5 ✓

Write them with x: (x + 2)(x + 3)

Signs Rule (Very Important!)

The sign of c (last term) tells you if the signs are the same or different:

If c is POSITIVE (+):
Signs are the SAME. Look at b to know which:
b positive → both +  |  b negative → both −

If c is NEGATIVE (−):
Signs are DIFFERENT. The larger number takes the sign of b.

Common Mistakes!

Mistake 1: Finding numbers that add to b, but forget to check if they multiply to c.

x² + 5x + 6 = (x + 1)(x + 4) × WRONG

1+4=5 but 1×4=4 ≠ 6!

x² + 5x + 6 = (x + 2)(x + 3) ✓ 2+3=5 and 2×3=6

Mistake 2: Getting the signs wrong.

x² − 5x + 6 = (x + 2)(x + 3) × WRONG

x² − 5x + 6 = (x − 2)(x − 3) ✓ c is +, b is − → both negative!

Mistake 3: Using DOTS when the middle term is not zero.

x² + 5x + 6 = (x + 6)(x − 1) × WRONG (that gives x²+5x−6)

x² + 5x + 6 = (x + 2)(x + 3) ✓

Worked Examples

Example 1: Factor: x² + 5x + 6
Need: × 6 and + 5
Factors of 6: (2, 3) → 2+3 = 5 ✓
→ c is +, b is + → both positive
→ (x + 2)(x + 3)

Example 2: Factor: x² − 7x + 12
Need: × 12 and − 7
Factors of 12: (−3, −4) → −3+( −4) = −7 ✓
→ c is +, b is − → both negative
→ (x − 3)(x − 4)

Example 3: Factor: x² + 2x − 8
Need: × −8 and + 2
Factors of −8: (+4, −2) → 4+( −2) = 2 ✓
→ c is − → different signs. Larger (4) takes sign of b (+)
→ (x + 4)(x − 2)

Example 4: Factor: x² − 3x − 10
Need: × −10 and − 3
Factors of −10: (−5, +2) → −5+2 = −3 ✓
→ c is − → different signs. Larger (5) takes sign of b (−)
→ (x − 5)(x + 2)

Example 5: Factor: x² + 9x + 20
Need: × 20 and + 9
Factors of 20: (4, 5) → 4+5 = 9 ✓
→ c is +, b is + → both positive
→ (x + 4)(x + 5)

Shortcut: List all factor pairs of c. Pick the pair that adds to b. Use the signs rule to place + or − correctly!

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 Quadratic Trinomials