If √(43 − 24√3) = a + b√3, where a and b are integers, then what is the value…

A nested surd of the form √(p − q√3) can be denested into a + b√3 by squaring both sides and matching the rational part with the coefficient of √3 separately — this works because 1 and √3 are linearly independent over the rationals. Because a square-root symbol always denotes the non-negative value, whichever integer pair solves the resulting two equations must still be checked so that a + b√3 itself comes out non-negative; if the check fails, the signs of both constants must be flipped.

1. Assume √(43 − 24√3) = a + b√3, with a and b integers as the wording specifies.

2. Square both sides: 43 − 24√3 = (a + b√3)2 = a2 + 3b2 + 2ab√3.

3. Match the rational parts: a2 + 3b2 = 43. Match the coefficients of √3: 2ab = −24, so ab = −12.

4. Substitute b = −12/a into a2 + 3b2 = 43 to get a4 − 43a2 + 432 = 0. Writing x = a2, this becomes x2 − 43x + 432 = 0, whose roots are x = (43 ± 11)/2 = 27 or 16. Since a must be an integer, only x = 16 gives an integer a (27 is not a perfect square), so a2 = 16.

5. a2 = 16 with ab = −12 gives two integer candidates: (a, b) = (4, −3) and (a, b) = (−4, 3).

6. Check each candidate against the non-negative requirement: 4 − 3√3 ≈ −1.196 is negative, so it is rejected; −4 + 3√3 ≈ 1.196 is positive, so (a, b) = (−4, 3) is the valid pair.

7. Compute 3a + 5b = 3(−4) + 5(3) = −12 + 15 = 3.

Squaring the valid pair confirms the work: (−4 + 3√3)2 = 16 − 24√3 + 27 = 43 − 24√3, exactly matching the original radicand, and −4 + 3√3 ≈ 1.196 is indeed non-negative. So a + b√3 = −4 + 3√3 is correctly √(43 − 24√3), and 3a + 5b = 3.