If (p/q - q/p) = 5/6, then find 4p/q + 4q/p ?

Concept: If a nonzero real number a satisfies a − 1/a = k, multiplying the equation through by a converts it into the quadratic a² − ka − 1 = 0. Solving this quadratic gives the value(s) of a, from which any expression in a + 1/a can then be evaluated directly.

Application:

1. Let a = p/q, so the given equation p/q − q/p = 5/6 becomes a − 1/a = 5/6.

2. Multiply through by a (a ≠ 0): a2 − 1 = (5/6)a.

3. Multiply by 6 and rearrange to clear fractions: 6a2 − 5a − 6 = 0.

4. Solve this quadratic (factor or use the quadratic formula): a = (5 ± 13)/12, giving a = 3/2 or a = −2/3.

5. Compute 4(a + 1/a) for each root: for a = 3/2, 1/a = 2/3, so a + 1/a = 3/2 + 2/3 = 13/6 and 4(a + 1/a) = 26/3; for a = −2/3, 1/a = −3/2, so a + 1/a = −13/6 and 4(a + 1/a) = −26/3.

6. Among the four offered options only 26/3 appears, so that is the value of 4p/q + 4q/p for this question.

Cross-check:

Substitute a = 3/2 back into the original relation: 3/2 − 2/3 = 9/6 − 4/6 = 5/6, which matches the given equation. Then 4(3/2) + 4(2/3) = 6 + 8/3 = 26/3, confirming the result.