Find the value of the expression: √(7 − 2√12)

CONCEPT: An expression of the form √(a − 2√b) can be simplified using the identity √(a − 2√b) = √p − √q, valid whenever positive numbers p and q (with p > q) satisfy p + q = a and p × q = b — this follows directly from expanding (√p − √q)² = p + q − 2√(pq).

APPLICATION:

1. Match the radicand 7 − 2√12 to the form a − 2√b: here a = 7 and 2√b = 2√12, so b = 12.

2. Find p and q with p + q = 7 and p × q = 12. Testing factor pairs of 12 — (1, 12), (2, 6), (3, 4) — the pair p = 4, q = 3 satisfies both: 4 + 3 = 7 and 4 × 3 = 12.

3. Apply the identity: √(7 − 2√12) = √4 − √3 = 2 − √3.

CROSS-CHECK: Squaring 2 − √3 gives (2 − √3)² = 4 − 4√3 + 3 = 7 − 4√3 = 7 − 2√12 (since 4√3 = 2 × 2√3 = 2√12), which reproduces the original radicand exactly, confirming the result.

So the value of the expression is 2 − √3.