Expand: (2 + √5)(3 - √5)
Solution
Use FOIL method:
= 2×3 + 2×(-√5) + √5×3 + √5×(-√5)
= 6 - 2√5 + 3√5 - 5
= 1 + √5
Answer: 1 + √5
This study notes covers Worked Example 5: Expanding with Surds within Surds for GCSE Mathematics. Revise Surds in Number for GCSE Mathematics with 14 exam-style questions and 22 flashcards. This is a high-frequency topic, so it is worth revising until the explanation feels precise and repeatable. It is section 11 of 15 in this topic. Use this study notes to connect the idea to the wider topic before moving on to questions and flashcards.
Practice questions for Surds
Which of these is the simplified form of √48?
Explain why it is preferable to write fractions in rationalized form rather than leaving a surd in the denominator.
Quick recall flashcards
What is a surd?
An irrational root that cannot be simplified to a whole number
Examples: √2, √3, √5, ∛7
NOT surds: √4 = 2, √9 = 3 (these simplify to whole numbers)
What are Like Surds?
Surds with the same root part
Examples of like surds:
• 3√2 and 5√2 (both have √2)
• 2√7 and -4√7 (both have √7)
Can add/subtract like surds:
3√2 + 5√2 = 8√2