What will come in place of ? to satisfy the equation : (√6−1)² =?−2√6

Correct — A.
Expand with (a − b)² = a² − 2ab + b², taking a = √6 and b = 1.

Square the first term: (√6)² = 6
Double the product: 2 × √6 × 1 = 2√6
Square the second term: 1² = 1

Collect: 6 − 2√6 + 1 = 7 − 2√6
Match with ? − 2√6: ? = 7 → option (a)

Why the others are wrong

• (b)
  6 — 6 is (√6)² on its own. It drops the +1 from squaring the 1: 6 − 2√6 ≈ 1.10, while (√6 − 1)² ≈ 2.10.
• (c)
  5 — 5 comes from 6 − 1, giving b² a minus sign. In (a − b)² only the middle term 2ab is subtracted; b² = 1 is added.
• (d)
  4 — 4 − 2√6 ≈ 4 − 4.90 is negative, but (√6 − 1)² is a square of a real number and can never be negative.

(a − b)² = a² − 2ab + b² works the same when a or b is a surd. Square each term, then add the cross term with its sign.

With a = √6 and b = 1, the two squares give whole numbers, 6 and 1. The surd survives only in the cross term 2ab = 2√6.

So the right side's whole-number part is 6 + 1 = 7 and its surd part is −2√6. Matching parts gives the blank directly.

Key facts

• (a − b)² = a² − 2ab + b², and (a + b)² = a² + 2ab + b²
• (√n)² = n for any n ≥ 0, so (√6)² = 6
• A square is never negative, so ? − 2√6 ≥ 0 and ? ≥ 2√6 ≈ 4.90

Common traps

• Squaring each term and dropping the cross term, which turns (√6 − 1)² into 6 + 1
• Subtracting b² as well as 2ab, which gives 5 − 2√6

No directly related past PYQ was found.