We will learn how to substitute numbers containing radicals for variables and find the value of an expression.
How to Find the Value of an Expression
Before substituting, it is efficient to simplify the expression by factoring or transforming it.
📘 Example ①
When x = √3 + 1, find the value of x² − 2x + 1.
Solution: x² − 2x + 1 = (x − 1)² = (√3 + 1 − 1)² = (√3)² = 3
When x = √3 + 1, find the value of x² − 2x + 1.
Solution: x² − 2x + 1 = (x − 1)² = (√3 + 1 − 1)² = (√3)² = 3
📘 Example ②
When a = √5 + 2 and b = √5 − 2, find the values of a + b and ab.
Solution: a + b = 2√5, ab = (√5)² − 2² = 5 − 4 = 1
When a = √5 + 2 and b = √5 − 2, find the values of a + b and ab.
Solution: a + b = 2√5, ab = (√5)² − 2² = 5 − 4 = 1
💡 Key Points
- First factor or transform the expression, then substitute to make the calculation easier
- Problems are often given using a pair consisting of a sum and a product
- Use multiplication formulas in reverse to make substitution easier
Practice Problems
- When x = √2 + 3, find the value of x² − 6x + 9.
- When a = √3 + 1 and b = √3 − 1, find the value of a² + b².
Answers and Explanations
- Answer: x² − 6x + 9 = (x − 3)² = (√2)² = 2
- Answer: a² + b² = (a+b)² − 2ab = (2√3)² − 2×(3−1) = 12 − 4 = 8