SSC CGL 2025 Tier-I · 16 September 2025, Shift 2 · Quantitative Aptitude · Question 2 of 100

We transcribed this text from the pictures on a candidate's response sheet for this shift; the answer is the key SSC colour-marked on that sheet. Solution last reviewed 13 September 2026.

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What is the value of:(√18 + √2)²−(√8 + √2)²

(a)14
(b)15
(c)16
(d)17

Answer

Why

Correct — A. Write each root as a multiple of √2, then square.

√18 = √(9 × 2) = 3√2, so √18 + √2 = 4√2
√8 = √(4 × 2) = 2√2, so √8 + √2 = 3√2

Square the first: (4√2)² = 16 × 2 = 32
Square the second: (3√2)² = 9 × 2 = 18
Subtract: 32 − 18 = 14 → option (a)

Why the others are wrong

(b)
15 — 15 is odd, but the two squares, 32 and 18, are both even, so their difference must be even. No correct working reaches 15.
(c)
16 — 16 would need (√8 + √2)² to be 16, that is √8 + √2 = 4. But √8 + √2 = 3√2 ≈ 4.24, and its square is 18.
(d)
17 — 17 is odd as well. Check each square by expansion: 18 + 2 + 2√36 = 32 and 8 + 2 + 2√16 = 18, so the gap is 14.

Concept

Simplify the roots before squaring. √18 and √8 both hide a √2: √18 = 3√2 and √8 = 2√2. Once each bracket is a multiple of √2, squaring is just (number)² × 2.

The direct route gives the same numbers. (√a + √b)² = a + b + 2√(ab), so (√18 + √2)² = 20 + 2√36 = 32 and (√8 + √2)² = 10 + 2√16 = 18.

Taking the brackets as a = √18 + √2 and b = √8 + √2, the identity a² − b² = (a − b)(a + b) gives a third route: a − b = √2 and a + b = 7√2, so the value is √2 × 7√2 = 14.

Key facts

√18 = 3√2 and √8 = 2√2.
(√a + √b)² = a + b + 2√(ab).
(k√2)² = 2k², so (4√2)² = 32 and (3√2)² = 18.

Study next

Simplifying surds by pulling out square factors, e.g. √50 = 5√2Difference of squares with surd termsComparing sums of surds by squaring both

Common traps

Squaring term by term: (√18 + √2)² is not 18 + 2. The cross term 2√36 = 12 is missing.
Adding under one root: √18 + √2 is 4√2, not √20.

The same √2 simplification decides 14 Sep 2025, 16:00, Quant Q.23, where √50 + √18 − √8 = 5√2 + 3√2 − 2√2 = 6√2.

At 12 Sep 2025, 12:30, Quant Q.1, (√8 + √2)² = 18 is what shows √8 + √2 is larger than √3 + √5, whose square is 8 + 2√15 ≈ 15.75.

Related PYQs

No directly related past PYQ was found.

Topic: Square roots — all 48 SSC CGL questions on it, every solved shift.