Introduction / Context:
This algebra question again uses the identity for the sum of cubes to relate the sum a + b, the sum of cubes a^3 + b^3, and the product ab. It is a classic pattern in aptitude exams where you are given symmetric expressions and must extract one unknown from the others.
Given Data / Assumptions:
- a^3 + b^3 = 35.
- ab = 6.
- a and b are real numbers.
- We must find a + b.
Concept / Approach:
We use the identity a^3 + b^3 = (a + b)^3 − 3ab(a + b). Let s = a + b. Then we can rewrite the given information as 35 = s^3 − 3ab s. Since ab = 6, this becomes a cubic equation in s. We then solve this cubic by simple trial of integer values from the options.
Step-by-Step Solution:
Let s = a + b.
Using the identity: a^3 + b^3 = s^3 − 3ab s.
Given a^3 + b^3 = 35 and ab = 6, substitute into the identity.
So 35 = s^3 − 3 * 6 * s = s^3 − 18s.
Rearrange: s^3 − 18s − 35 = 0.
Now test simple integer values. For s = 5: 5^3 − 18 * 5 − 35 = 125 − 90 − 35 = 0.
Since s = 5 satisfies the cubic, we have a + b = 5.
Verification / Alternative check:
We can confirm consistency by constructing a quadratic with sum 5 and product 6: t^2 − 5t + 6 = 0 with roots t = 2 and t = 3. For a = 2 and b = 3, we have a + b = 5 and a^3 + b^3 = 8 + 27 = 35, perfectly matching the given data.
Why Other Options Are Wrong:
Options b, c, and d do not satisfy the cubic equation s^3 − 18s − 35 = 0. Substituting s = 8, 2, or −8 does not yield zero for this expression. Option e (−5) also fails. Only s = 5 makes the identity consistent with the given numbers.
Common Pitfalls:
Students sometimes forget that a^3 + b^3 can be expressed in terms of a + b and ab, or they attempt to guess a and b directly without using the identity, which can be time consuming. Using the identity systematically is far more efficient and reliable in exam settings.
Final Answer:
The sum of the two numbers is
a + b = 5.
Discussion & Comments