If x = √2 + 1, find x²−2.
- (a)1 + 2√2
- (b)3 + 2√2
- (c)2 + √2
- (d)1 + √2
Answer
Why
Correct — A.
Expand with (a + b)² = a² + 2ab + b²:
x² = (√2)² + 2 × √2 × 1 + 1²
= 2 + 2√2 + 1 = 3 + 2√2
Subtract 2: x² − 2 = 3 + 2√2 − 2
= 1 + 2√2 → option (a)
Why the others are wrong
- (b)3 + 2√2 — 3 + 2√2 is x² itself: the square is right, but the final step, subtract 2, is left undone.
- (c)2 + √2 — 2 + √2 ≈ 3.41, while x² − 2 ≈ 5.83 − 2 = 3.83. It also lacks the 2√2 cross-term that squaring √2 + 1 produces.
- (d)1 + √2 — 1 + √2 is x itself, the value given in the question. Nothing has been squared or subtracted.
Concept
Squaring a two-term surd uses (a + b)² = a² + 2ab + b². The cross-term 2ab is where the surd survives: (√2 + 1)² = 3 + 2√2, while the rational parts 2 and 1 combine.
A second route: x = 1 + √2 means x − 1 = √2. Squaring gives x² − 2x + 1 = 2, so x² = 2x + 1. Then x² − 2 = 2x − 1 = 2 + 2√2 − 1 = 1 + 2√2.
A decimal check separates the options fast. With √2 ≈ 1.414, x ≈ 2.414 and x² − 2 ≈ 3.83. The options come to 3.83, 5.83, 3.41 and 2.41, and 1 + 2√2 is the one that matches.
Key facts
- (a + b)² = a² + 2ab + b²
- (√2 + 1)² = 3 + 2√2
- (√2 + 1)(√2 − 1) = 1, so 1 ⁄ (√2 + 1) = √2 − 1
Study next
Common traps
- Writing (√2 + 1)² as 2 + 1 = 3 and losing the 2√2 cross-term.
- Stopping at x² = 3 + 2√2, which is printed among the options, without subtracting 2.
A surd substituted into an expression is also set at 20 Sep 2025, 09:00, Quant Q.21 (x = 2 + √3, find x − 1⁄x, keyed 2√3) and at 26 Sep 2024, 12:30, Quant Q.6 (x = √6 + 2 and y = √6 − 2, where the cross-terms of x² and y² cancel).
Related PYQs
No directly related past PYQ was found.