Let us combine the formulas learned so far to expand more complicated expressions and calculate the values of expressions using expansion.
Expansions using two or more formulas
When several products appear in the same expression, expand each one and then combine like terms.
Example: (x + 3)² − (x + 1)(x − 4)
= (x² + 6x + 9) − (x² − 3x − 4)
= x² + 6x + 9 − x² + 3x + 4 = 9x + 13
Expand (2x − 1)(2x + 1) − (x − 3)².
Answer:(4x² − 1) − (x² − 6x + 9) = 4x² − 1 − x² + 6x − 9 = 3x² + 6x − 10
Replacing a common part with a variable
Example: For (a + b + 3)(a + b − 3), let a + b = A. Then (A + 3)(A − 3) = A² − 9 = (a + b)² − 9 = a² + 2ab + b² − 9.
Expand (x + y − 4)².
Answer:Let x + y = A. Then (A − 4)² = A² − 8A + 16 = (x+y)² − 8(x+y) + 16 = x² + 2xy + y² − 8x − 8y + 16
Calculating the value of an expression
When the values are complicated, such as x = 5+√3 and y = 5−√3, it is easier to first transform the expression using expansion or factoring and then substitute the values.
Example: When x = 6+√2 and y = 6−√2, xy = (6+√2)(6−√2) = 36 − 2 = 34, and x+y = 12.
x² + y² = (x+y)² − 2xy = 144 − 68 = 76.
- When there are multiple terms, expand each part and then combine like terms
- If there is a common part, replacing it with one variable allows you to use a formula
- Remember transformations such as x² + y² = (x+y)² − 2xy and (x−y)² = (x+y)² − 4xy
- Checking after expansion: substitute a simple value for x and confirm that both sides agree
Practice Problems
- Expand (x + 5)² − (x + 2)(x − 2).
- Expand (a + b + 1)(a + b − 1).
- When x = 3+√5 and y = 3−√5, find the value of x² + y².
Answers and Explanations
- Answer:10x + 29
Explanation: (x²+10x+25) − (x²−4) = 10x + 29。 - Answer:a² + 2ab + b² − 1
Explanation: Let a+b=A. Then (A+1)(A−1) = A² − 1 = (a+b)² − 1。 - Answer:28
Explanation: x+y=6、xy=9−5=4。x²+y² = (x+y)² − 2xy = 36 − 8 = 28。