What will come in place of ? to satisfy the equation : (√3 + 2)² =? + 4√3

Correct — A. Expand the square with (a + b)² = a² + 2ab + b².

a² = (√3)² = 3
2ab = 2 × √3 × 2 = 4√3
b² = 2² = 4

Add the parts: 3 + 4 + 4√3 = 7 + 4√3
Match with ? + 4√3: the surd parts agree, so ? = 7 → option (a)

Why the others are wrong

• (b)
  6 — 6 + 4√3 is one less than the true square, 7 + 4√3. The rational part is (√3)² + 2² = 3 + 4 = 7, so ? must be 7.
• (c)
  5 — 5 would need (√3)² + 2² to equal 5. But 2² is 4, not 2, so the rational part is 3 + 4 = 7.
• (d)
  4 — 4 is b² = 2² alone. (√3)² = 3 is a whole number too, so it joins the rational part: 3 + 4 = 7.

Squaring a binomial that holds a surd gives two kinds of term: rational ones from the two squares and a surd term from the middle.

(√3 + 2)² = (√3)² + 2 × √3 × 2 + 2² = 3 + 4√3 + 4.

Two numbers p + q√3 and r + s√3, with p, q, r, s rational, are equal exactly when p = r and q = s. The 4√3 matches on both sides, so ? is the rational part, 7.

Key facts

• (a + b)² = a² + 2ab + b².
• (√3)² = 3: squaring a square root returns the number under it.
• If p + q√3 = r + s√3 with p, q, r, s rational, then p = r and q = s.

Common traps

• Writing the middle term as 2√3 instead of 2 × √3 × 2 = 4√3.
• Adding 2 instead of 2² = 4 for the last term, which gives 5.

No directly related past PYQ was found.