Factoring ② Formula 1 (Product of a Sum and a Difference)
Using the expansion formula (a+b)(a−b) = a² − b² in reverse, we can factor a² − b² as a² − b² = (a+b)(a−b).
Formula for the product of a sum and a difference
a² − b² = (a + b)(a − b)
The difference of two squares can be factored into the product of their sum and difference.
📘 Example ①
Factor x² − 9.
Answer:x² − 9 = x² − 3² = (x + 3)(x − 3)
Factor x² − 9.
Answer:x² − 9 = x² − 3² = (x + 3)(x − 3)
📘 Example ②
Factor 4a² − 25.
Answer:4a² − 25 = (2a)² − 5² = (2a + 5)(2a − 5)
Factor 4a² − 25.
Answer:4a² − 25 = (2a)² − 5² = (2a + 5)(2a − 5)
💡 Key points
- Check whether the two terms are each in square form
- Rewrite the expression in square form including the coefficient（for example: 4x² = (2x)²）
- If there is a common factor, take it out before using the formula
Practice Problems
- Factor x² − 16.
- Factor 9x² − 4.
Answers and Explanations
- Answer:x² − 16 = x² − 4² = (x + 4)(x − 4)
- Answer:9x² − 4 = (3x)² − 2² = (3x + 2)(3x − 2)