Core Concept
Why Factor?
If A × B = 0, then either A = 0 or B = 0. This is the zero product property. Factoring rewrites the quadratic as a product of two brackets, then we set each bracket to zero.
x² + 5x + 6 = 0
(x + 2)(x + 3) = 0
x = −2 or x = −3
Method
Factoring When a = 1
For x² + bx + c, find two numbers that multiply to c and add to b.
Example: Solve x² + 7x + 12 = 0
1. Need two numbers: multiply to 12, add to 7
2. Pairs that multiply to 12: (1,12), (2,6), (3,4)
3. 3 + 4 = 7 ✓
4. Factor: (x + 3)(x + 4) = 0
5. x = −3 or x = −4
Negative Numbers
Factoring with Negatives
Example: Solve x² − x − 12 = 0
1. Need: multiply to −12, add to −1
2. Try: (−4)(3) = −12, and −4 + 3 = −1 ✓
3. Factor: (x − 4)(x + 3) = 0
4. x = 4 or x = −3
Special Cases
Difference of Squares
When you have x² − k² (a perfect square minus a perfect square), it always factors as:
x² − k² = (x + k)(x − k)
Examples:
x² − 25 = (x + 5)(x − 5) → x = ±5
x² − 49 = (x + 7)(x − 7) → x = ±7
4x² − 9 = (2x + 3)(2x − 3) → x = ±3/2
Advanced
Factoring When a ≠ 1
Example: Factor 2x² + 7x + 3
1. Multiply a × c: 2 × 3 = 6
2. Find two numbers: multiply to 6, add to 7 → (1, 6)
3. Rewrite middle: 2x² + 1x + 6x + 3
4. Group: x(2x + 1) + 3(2x + 1)
5. Factor: (x + 3)(2x + 1)
6. x = −3 or x = −1/2
Watch Out
Common Mistakes
- Forgetting to set each bracket equal to zero after factoring
- Sign errors: if c is negative, the brackets have opposite signs
- Forgetting to factor out a common factor first (e.g., 2x² + 4x = 2x(x+2))
- Always check your factoring by expanding back out
Your Turn
Try It Yourself
Q1. Solve x² + 6x + 8 = 0 by factoring.
Show Answer
(x + 2)(x + 4) = 0 → x = −2 or x = −4
Q2. Factor and solve x² − 16 = 0.
Show Answer
(x + 4)(x − 4) = 0 → x = ±4
Q3. Solve x² − 3x − 10 = 0.
Show Answer
Need: multiply to −10, add to −3 → (−5)(2)
(x − 5)(x + 2) = 0 → x = 5 or x = −2
Key Takeaways
1-Minute Summary
- Find two numbers that multiply to c and add to b
- Zero product property: if (x+p)(x+q)=0, then x=−p or x=−q
- Difference of squares: x²−k² = (x+k)(x−k)
- Always check by expanding your answer