Simplify the expression:√(9 + 4√5)
- (a)√5 + 2
- (b)3 + √5
- (c)2 + √6
- (d)3 + √4
Answer
Why
Correct — A.
Aim: write 9 + 4√5 as a square, using (√p + √q)² = p + q + 2√(pq).
Rewrite the surd with a 2 in front: 4√5 = 2√20, so p + q = 9 and pq = 20
Find the pair: 5 + 4 = 9 and 5 × 4 = 20
Square to check: (√5 + √4)² = 5 + 4 + 2√20 = 9 + 4√5
Take the root: √(9 + 4√5) = √5 + √4 = √5 + 2 → option (a)
Why the others are wrong
- (b)3 + √5 — (3 + √5)² = 14 + 6√5, not 9 + 4√5. In decimals 3 + √5 ≈ 5.24, while √(9 + 4√5) ≈ 4.24.
- (c)2 + √6 — (2 + √6)² = 10 + 4√6. The surd left over is √6, so this can never rebuild 9 + 4√5.
- (d)3 + √4 — √4 = 2, so 3 + √4 is just 5, and 5² = 25, not 9 + 4√5 ≈ 17.94. The √5 has disappeared altogether.
Concept
Denesting a square root: if a + 2√b can be written as (√p + √q)², its root is √p + √q. That square expands to p + q + 2√(pq).
So look for two numbers with sum a and product b. First force the coefficient of the surd to be 2: here 4√5 = 2√20, so you need sum 9 and product 20, which 5 and 4 give.
For a − 2√b the root is √p − √q with p > q, so the result stays positive.
A decimal check is quick: 9 + 4√5 ≈ 17.94, whose root is about 4.24. The four options come to about 4.24, 5.24, 4.45 and 5, so only √5 + 2 matches.
Key facts
- (√p + √q)² = p + q + 2√(pq)
- √(a + 2√b) = √p + √q when p + q = a and pq = b
- Move the surd's coefficient to 2 first: 4√5 = 2√20, 6√2 = 2√18
- √5 ≈ 2.236, so √5 + 2 ≈ 4.236
Study next
Common traps
- Looking for a pair with sum 9 and product 5 by reading 4√5 as if it were 2√5
- Squaring √5 + 2 as 5 + 4 and dropping the cross term 2√(pq)
21 Sep 2025, 16:00, Quant Q.24 applies the same split to a quotient: √(5 + 2√6) = √3 + √2 and √(5 − 2√6) = √3 − √2, and the answer is keyed 5 + 2√6.
Related PYQs
No directly related past PYQ was found.