If x = √((2 + √3)⁄(2 − √3)), then determine the value of x² + x − 9?

Correct — B. Rationalise inside the root, then substitute.

Multiply top and bottom by 2 + √3: the denominator becomes (2 − √3)(2 + √3) = 4 − 3 = 1
So the fraction is (2 + √3)², and x = 2 + √3
Square it: x² = 4 + 4√3 + 3 = 7 + 4√3

x² + x − 9 = 7 + 4√3 + 2 + √3 − 9
= (7 + 2 − 9) + (4√3 + √3) = 5√3 → option (b)

Why the others are wrong

• (a)
  3√3 — 3√3 ≈ 5.20 fails the decimal check: x ≈ 3.732, x² ≈ 13.928, and 13.928 + 3.732 − 9 ≈ 8.660, which is 5√3.
• (c)
  7√3 — 7 is the rational part of x² = 7 + 4√3, not a √3 coefficient. The √3 terms are 4√3 from x² and √3 from x, 5√3 in all.
• (d)
  9√3 — The −9 cancels the rational parts 7 + 2 exactly. It never multiplies √3, so nothing in the working produces 9√3.

Rationalising (a + √b)⁄(a − √b) means multiplying top and bottom by the conjugate a + √b. The denominator becomes a² − b.

Here a² − b = 4 − 3 = 1, so (2 + √3)⁄(2 − √3) is simply (2 + √3)², and its square root is 2 + √3.

Because (2 + √3)(2 − √3) = 1, the two numbers are reciprocals, and x + 1⁄x = 4. That gives a second route: x² = 4x − 1, so x² + x − 9 = 5x − 10 = 5√3.

Key facts

• (2 + √3)(2 − √3) = 4 − 3 = 1, so 2 − √3 = 1⁄(2 + √3).
• (2 + √3)² = 7 + 4√3.
• x = 2 + √3 satisfies x + 1⁄x = 4, that is, x² − 4x + 1 = 0.
• 2 + √3 ≈ 3.732.

Common traps

• Multiplying by (2 − √3)⁄(2 − √3) is legal but gives 1⁄(7 − 4√3), which still needs rationalising. Use the conjugate of the denominator, 2 + √3.
• Forgetting the square root. The fraction equals (2 + √3)², so x is 2 + √3, not 7 + 4√3.

No directly related past PYQ was found.