There are formulas that must be memorized in order to expand expressions quickly.
Quadratic Multiplication Formulas
- (a + b)² = a² + 2ab + b²
- (a − b)² = a² − 2ab + b²
- (a + b)(a − b) = a² − b²
- (x + a)(x + b) = x² + (a + b)x + ab
- (ax + b)(cx + d) = acx² + (ad + bc)x + bd
Example ①
(1) (x + 5)² (2) (2x − 3)² (3) (x + 4)(x − 4)
Answer: (1) x² + 10x + 25 (2) 4x² − 12x + 9 (3) x² − 16
(1) (x + 5)² (2) (2x − 3)² (3) (x + 4)(x − 4)
Answer: (1) x² + 10x + 25 (2) 4x² − 12x + 9 (3) x² − 16
Example ②
(1) (x + 3)(x − 7) (2) (2x + 5)(3x − 4)
Answer: (1) x² − 4x − 21 (2) 6x² + 7x − 20
(1) (x + 3)(x − 7) (2) (2x + 5)(3x − 4)
Answer: (1) x² − 4x − 21 (2) 6x² + 7x − 20
Key Point
- Be careful with the sign of the middle term in (a+b)² and (a−b)²
- (a+b)(a−b) = a² − b² is the product of a sum and a difference
- In the cross-multiplication expansion (ax+b)(cx+d), ad+bc is the middle term
Practice Problems
- Expand (3x − 2)².
- Expand (x + 6)(x − 6).
- Expand (2x + 1)(5x − 3).
Answers and Explanations
- Answer: 9x² − 12x + 4
Explanation: (3x)² − 2·3x·2 + 2² = 9x² − 12x + 4. - Answer: x² − 36
Explanation: Use the product of a sum and a difference: x² − 6². - Answer: 10x² − x − 3
Explanation: 10x² + (−6+5)x − 3.