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FACTORING: QUADRATIC TRINOMIAL

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!

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 Quadratic Trinomials. You got this!

Answer all questions first before submitting!
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