The sum of the squares of two numbers is 234, and the square of their difference is 144. Find the product of the two numbers.
Aptitude
Elementary Algebra
Difficulty: Easy
Choose an option
-
A28
-
B52
-
C36
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D45
-
E40
Answer
Correct Answer: 45
Explanation
Introduction / Context:This problem uses algebraic identities that relate sums and differences to products. Specifically, knowing a^2 + b^2 and (a − b)^2 enables solving for ab via identity manipulations without finding a and b individually.
Given Data / Assumptions:
- a^2 + b^2 = 234.
- (a − b)^2 = 144.
- Need ab.
Concept / Approach:Use the identity (a − b)^2 = a^2 + b^2 − 2ab. Rearranging gives 2ab = (a^2 + b^2) − (a − b)^2. Substitute the given values to compute ab directly.
Step-by-Step Solution:
(a − b)^2 = a^2 + b^2 − 2ab144 = 234 − 2ab2ab = 234 − 144 = 90ab = 90 / 2 = 45Verification / Alternative check:If desired, we can choose numbers with product 45 that fit the constraints (not necessary here). The identity-based computation is sufficient and exact.
Why Other Options Are Wrong:
- 28, 52, 36, 40: These do not satisfy 2ab = 90; substituting back fails the identity relation.
Common Pitfalls:
- Using (a + b)^2 identity mistakenly instead of (a − b)^2.
- Sign errors when isolating 2ab.
Final Answer:45