We summarize what we have learned so far about applications of square roots and work on comprehensive problems.
Summary
The following points are important when applying square roots.
・Value of an expression: Transform the expression by factoring before substitution
・Integer part and decimal part: Identify the integer part using the order relationship
・Using multiplication formulas: Combine the radical terms after expansion
・Using approximate values: Organize the expression before substitution
📘 Example ① (Comprehensive)
When x = √2, find the value of x² + 4x + 4.
Answer:x² + 4x + 4 = (x + 2)² = (√2 + 2)² = 2 + 4√2 + 4 = 6 + 4√2
When x = √2, find the value of x² + 4x + 4.
Answer:x² + 4x + 4 = (x + 2)² = (√2 + 2)² = 2 + 4√2 + 4 = 6 + 4√2
📘 Example ② (Comprehensive)
Calculate (√3 + √2)² − (√3 − √2)².
Answer:Expanding gives (3+2√6+2) − (3−2√6+2) = 5+2√6 − 5+2√6 = 4√6
Calculate (√3 + √2)² − (√3 − √2)².
Answer:Expanding gives (3+2√6+2) − (3−2√6+2) = 5+2√6 − 5+2√6 = 4√6
💡 Key Points
- For a complicated expression, consider whether it can be transformed using A² − B² = (A+B)(A−B)
- After calculating, organize the radical part and the rational-number part separately
- When checking your answer, verify it by expanding or factoring
Practice Problems
- When x = 2√3, find the value of x² − 12.
- Calculate (√5 + 1)² − (√5 − 1)².
Answers and Explanations
- Answer:x² − 12 = (2√3)² − 12 = 12 − 12 = 0
- Answer:(6+2√5) − (6−2√5) = 4√5