If three real numbers a, b and c satisfy a^2 + b^2 + c^2 = 83 and a + b + c = 15, what is the exact value of the symmetric sum ab + bc + ca?

Introduction / Context:
This question checks understanding of symmetric sums and the identity relating the square of a sum to the sum of squares and pairwise products. Such identities are very useful in algebra and in many aptitude exams to find expressions like ab + bc + ca without solving individually for a, b, and c.

Given Data / Assumptions:

• a^2 + b^2 + c^2 = 83.
• a + b + c = 15.
• a, b, c are real numbers.
• We must find ab + bc + ca.

Concept / Approach:
Use the identity for the square of a sum of three numbers: (a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca). This equation relates the given sum and the sum of squares to the unknown symmetric sum ab + bc + ca. Rearranging the identity makes it easy to solve for ab + bc + ca directly.

Step-by-Step Solution:
Start with the identity (a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca).Substitute the given values: (15)^2 = 83 + 2(ab + bc + ca).Compute 15^2: 225 = 83 + 2(ab + bc + ca).Subtract 83 from both sides: 225 − 83 = 2(ab + bc + ca).This gives 142 = 2(ab + bc + ca), so ab + bc + ca = 142 / 2 = 71.

Verification / Alternative check:
If a, b, and c are roots of some cubic equation, their symmetric sums would appear in the coefficients. The identity used is a standard result and holds for all real a, b, and c. Substituting back, we see that a^2 + b^2 + c^2 + 2(ab + bc + ca) = 83 + 2 * 71 = 83 + 142 = 225, which is exactly (a + b + c)^2 = 15^2. This confirms that the calculated value of 71 is consistent with the given data.

Why Other Options Are Wrong:

• 69, 70, 72, and 68 do not satisfy the identity when substituted back, because they give different totals on the right side of (a + b + c)^2.
• For example, using 70 would give 83 + 2 * 70 = 223, which is not equal to 225.
• Only 71 leads to the correct equality 83 + 2 * 71 = 225.

Common Pitfalls:

• Forgetting the factor of 2 in the identity or writing (a + b + c)^2 = a^2 + b^2 + c^2 + ab + bc + ca, which is incorrect.
• Making arithmetic errors when computing 225 − 83.
• Attempting to solve for individual values of a, b, and c instead of using the symmetric identity, which is unnecessary and complex.

Final Answer:
71