If x + y = 10 and x^2 + y^2 = 68, then what is the value of the product xy?

Introduction / Context:
This algebra question uses relationships between sums and squares of two numbers. Problems of this type are common in aptitude tests and are based on the identity that connects x + y, x^2 + y^2 and the product xy. The goal is to find xy without solving individually for x and y.

Given Data / Assumptions:

• x + y = 10
• x^2 + y^2 = 68
• x and y are real numbers
• We need the value of xy.

Concept / Approach:
We rely on the standard identity (x + y)^2 = x^2 + 2xy + y^2. If we know the sum of the numbers and the sum of their squares, we can substitute these into the identity and then solve for xy. This method avoids solving simultaneous equations directly and is much faster in an exam setting.

Step-by-Step Solution:
Step 1: Start from the identity (x + y)^2 = x^2 + 2xy + y^2. Step 2: Substitute x + y = 10, so (x + y)^2 = 10^2 = 100. Step 3: Substitute x^2 + y^2 = 68 into the identity. Step 4: The equation becomes 100 = 68 + 2xy. Step 5: Rearrange to find 2xy = 100 - 68 = 32. Step 6: Divide both sides by 2 to get xy = 16.

Verification / Alternative check:
We can check by trying to find numbers that satisfy both conditions. Suppose x = 8 and y = 2. Then x + y = 10 and x^2 + y^2 = 64 + 4 = 68, which works. The product xy is 8 * 2 = 16, confirming our derived value.

Why Other Options Are Wrong:

• Option A: 21 would correspond to 2xy = 42, which does not satisfy the identity with the given numbers.
• Option B: 24 would give 2xy = 48, leading to x^2 + y^2 = 52 instead of 68.
• Option C: 25 would produce 2xy = 50 and x^2 + y^2 = 50, which contradicts the given 68.
• Option E: 18 also leads to the wrong total when substituted into the identity.

Common Pitfalls:
Students sometimes confuse the identity for (x - y)^2 with the one for (x + y)^2, or forget the factor of 2 in front of xy. Others try to solve for x and y individually, which takes longer and can introduce additional errors. Using the identity directly is both efficient and reliable.

Final Answer:
The required product is xy = 16.