Positive numbers a and b, with a > b, satisfy a² + b² = 6ab. What is the value of (a³ + b³)/(a³ − b³)?
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Answer & Explanation
Correct answer
B
Don't solve for a and b individually, since the relation gives a/b = 3 + 2√2, an ugly irrational that makes cubing intractable. Instead read a² + b² = 6ab as a tool for building the symmetric pairs (a + b)² and (a − b)².
Add 2ab to both sides: (a + b)² = a² + 2ab + b² = 6ab + 2ab = 8ab. Subtract 2ab: (a − b)² = 6ab − 2ab = 4ab. Dividing, (a + b)²/(a − b)² = 8ab/4ab = 2, and since a is greater than b with both positive, (a + b)/(a − b) = √2.
Now factor the cubes using the same relation. The sum of cubes is (a + b)(a² − ab + b²), and since a² + b² = 6ab the second factor is 6ab − ab = 5ab. The difference of cubes is (a − b)(a² + ab + b²) = (a − b)(6ab + ab) = (a − b)(7ab). So the ratio is (a + b)(5ab) / [(a − b)(7ab)]. The ab cancels, leaving (5/7) × (a + b)/(a − b) = (5/7)√2, that is 5√2/7. Numerical check with b = 1, a ≈ 5.828: the ratio is about 1.010, matching 5√2/7 ≈ 1.010. So the answer is B.
The distractors come from identity and surd slips. 7√2/5 (E) inverts the 5-to-7 coefficient ratio by mis-assigning which factored form carries the +ab versus −ab middle term. 10/7 (D) reads off (a + b)²/(a − b)² = 2 and forgets to take the square root, using 2 instead of √2. The bare √2 (C) cancels a² − ab + b² against a² + ab + b² as if they were equal, discarding the 5-to-7 ratio. 5/7 (A) is the mirror error, keeping the 5/7 ratio but dropping the surd, treating (a + b)/(a − b) as 1.