Calculating four terms every time is difficult, so memorize frequently occurring forms as formulas. The first formula is as follows.
Formula ①: (x + a)(x + b) = x² + (a + b)x + ab
This is a simple rule: use the sum of the two constants as the coefficient of x, and their product as the constant term.
- (x + 2)(x + 5) = x² + (2+5)x + 2×5 = x² + 7x + 10
- (x − 3)(x + 7) = x² + (−3+7)x + (−3)×7 = x² + 4x − 21
- (x − 4)(x − 6) = x² + (−4−6)x + (−4)×(−6) = x² − 10x + 24
Example ①
Expand the following.
(1) (x + 6)(x − 2) (2) (x − 8)(x − 3)
Answer:
(1) Sum 6+(−2)=4, product 6×(−2)=−12 → x² + 4x − 12
(2) Sum −8+(−3)=−11, product (−8)×(−3)=24 → x² − 11x + 24
Expand the following.
(1) (x + 6)(x − 2) (2) (x − 8)(x − 3)
Answer:
(1) Sum 6+(−2)=4, product 6×(−2)=−12 → x² + 4x − 12
(2) Sum −8+(−3)=−11, product (−8)×(−3)=24 → x² − 11x + 24
The Same Applies When There Are Two Variables
Even if x in (x + a)(x + b) is another variable, or a and b are variables, the same formula can be used.
(x + 2y)(x + 3y) = x² + (2y + 3y)x + 2y×3y = x² + 5xy + 6y²
Example ②
Expand (a − 4b)(a + 9b).
Answer: Sum −4b+9b=5b, product (−4b)×9b=−36b² → a² + 5ab − 36b²
Expand (a − 4b)(a + 9b).
Answer: Sum −4b+9b=5b, product (−4b)×9b=−36b² → a² + 5ab − 36b²
Key Points
- (x+a)(x+b) = x² + (a+b)x + ab
- The coefficient of x is the “sum of the two numbers,” and the constant term is their “product.”
- Include the signs when identifying a and b. For example, in (x−3), a=−3.
- Whether x is another variable or a and b are variables, use the same formula if the form is the same.
Practice Problems
- Expand (x + 4)(x + 9).
- Expand (x − 5)(x + 8).
- Expand (a − 7b)(a − 2b).
Answer and Explanation
- Answer: x² + 13x + 36
Explanation: Sum 4+9=13, product 4×9=36. - Answer: x² + 3x − 40
Explanation: Sum −5+8=3, product (−5)×8=−40. - Answer: a² − 9ab + 14b²
Explanation: Sum −7b+(−2b)=−9b, product (−7b)×(−2b)=14b².