Another frequently used formula is the “product of a sum and a difference.” When expanded, the middle terms cancel, giving a simple form.
Formula ④: (x + a)(x − a) = x² − a²
When you multiply the sum and difference of the same x and a, the middle terms cancel each other, resulting in x² − a² (the difference of two squares).
- (x + 6)(x − 6) = x² − 6² = x² − 36
- (3x + 2)(3x − 2) = (3x)² − 2² = 9x² − 4
- (5a + 4b)(5a − 4b) = (5a)² − (4b)² = 25a² − 16b²
Example ①
Expand the following.
(1) (x + 10)(x − 10) (2) (2a − 7)(2a + 7)
Answer:
(1) x² − 10² = x² − 100
(2) (2a)² − 7² = 4a² − 49 （The order may be reversed with the same result）
Expand the following.
(1) (x + 10)(x − 10) (2) (2a − 7)(2a + 7)
Answer:
(1) x² − 10² = x² − 100
(2) (2a)² − 7² = 4a² − 49 （The order may be reversed with the same result）
Efficient numerical calculation using the formula
Using (x+a)(x−a) = x²−a² makes mental calculation easier.
Example: 103 × 97 = (100+3)(100−3) = 100² − 3² = 10000 − 9 = 9991
Example ②
Calculate 52 × 48 using the formula.
Answer:52 × 48 = (50+2)(50−2) = 50² − 2² = 2500 − 4 = 2496
Calculate 52 × 48 using the formula.
Answer:52 × 48 = (50+2)(50−2) = 50² − 2² = 2500 − 4 = 2496
Key points
- (x+a)(x−a) = x² − a²（the difference of two squares）
- The order may be reversed（(x−a)(x+a) gives the same result）
- Square the entire term even when it contains coefficients or variables
- Mental calculation method: using (a+b)(a−b) = a²−b² is quick
Practice Problems
- Expand (x + 9)(x − 9).
- Expand (4y − 3)(4y + 3).
- Calculate 98 × 102 using the formula.
Answers and Explanations
- Answer:x² − 81
Explanation: x² − 9² = x² − 81。 - Answer:16y² − 9
Explanation: (4y)² − 3² = 16y² − 9。 - Answer:9996
Explanation: (100−2)(100+2) = 100² − 4 = 10000 − 4 = 9996。