Which of the following equals 10?
- (a)(√7 + √3)² − 2√21
- (b)(√5 + √5)²
- (c)(√2 + √3)² + 2√6
- (d)(√11 + √5)² − 2√55
Answer
Why
Correct — A. Expand the square, then cancel the surd.
Rule: (√x + √y)² = x + y + 2√(xy)
Square: (√7 + √3)² = 7 + 3 + 2√21 = 10 + 2√21
Subtract 2√21: 10 + 2√21 − 2√21
= 10 → option (a)
Why the others are wrong
- (b)(√5 + √5)² — Dropping the cross term makes (b) look like 5 + 5 = 10. In full, (√5 + √5)² = 5 + 5 + 2√25 = 20, the same as (2√5)² = 4 × 5.
- (c)(√2 + √3)² + 2√6 — Here the cross term is added, not cancelled. (√2 + √3)² = 5 + 2√6, and a further 2√6 gives 5 + 4√6 ≈ 14.8, not even a whole number.
- (d)(√11 + √5)² − 2√55 — The surd cancels, but 11 + 5 = 16. Subtracting 2√55 leaves just the two numbers under the roots, and they add to 16, not 10.
Concept
(√x + √y)² = x + y + 2√(xy). The two squares give whole numbers, and the cross term carries the surd.
Subtract exactly 2√(xy) and the surd vanishes, leaving x + y. So (√7 + √3)² − 2√21 = 7 + 3 = 10, and the question becomes: which pair of numbers under the roots adds to 10?
Options (a) and (d) have the same shape and differ only in their numbers: 7 + 3 = 10 against 11 + 5 = 16.
Forgetting the cross term turns option (b) into 5 + 5 = 10, which is why the square is worth expanding in full.
Key facts
- (√x + √y)² = x + y + 2√(xy)
- (√x − √y)² = x + y − 2√(xy)
- √5 + √5 = 2√5, so (√5 + √5)² = 4 × 5 = 20.
Study next
Common traps
- Squaring a sum term by term, so that (√5 + √5)² looks like 5 + 5.
- Cancelling the surd in option (c) as in option (a), when its sign is + and the two 2√6 terms add.
The same expansion is asked at 21 Sep 2025, 16:00, Quant Q.1: (√6 − 1)² = 7 − 2√6, so the blank is 7.
21 Sep 2025, 16:00, Quant Q.24 runs it backwards: 5 + 2√6 = (√3 + √2)², which turns √(5 + 2√6)⁄√(5 − 2√6) into 5 + 2√6.
Related PYQs
No directly related past PYQ was found.