If a^2 + b^2 + c^2 = 14 and a + b + c = 6, then what is the value of the sum ab + bc + ca?

Introduction / Context:

This algebra problem tests your familiarity with the identity for the square of a sum of three numbers. It is a standard type seen in aptitude exams and algebra practice, where you use one equation involving squares and another involving simple sums to find mixed products.

Given Data / Assumptions:

• a^2 + b^2 + c^2 = 14.
• a + b + c = 6.
• We need to find ab + bc + ca.
• a, b, and c are real numbers.

Concept / Approach:

Use the identity for the square of a sum of three variables: (a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca). The values of a + b + c and a^2 + b^2 + c^2 are known, so we can substitute them and solve for ab + bc + ca.

Step-by-Step Solution:

Verification / Alternative check:

If you wish, you can imagine particular values of a, b, and c that satisfy both conditions and then check that ab + bc + ca equals 11. However, the identity based method is general and does not require guessing.

Why Other Options Are Wrong:

The values 12, 13, 14, and 10 come from incorrect arithmetic, such as using 36 + 14 instead of 36 - 14, or dividing by 3 instead of 2. Only 11 satisfies the identity for the given data.

Common Pitfalls:

One common error is to misremember the identity and write (a + b + c)^2 = a^2 + b^2 + c^2 + ab + bc + ca, forgetting the factor 2. Another mistake is to treat ab + bc + ca as (a + b + c)^2 - (a^2 + b^2 + c^2) without dividing by 2.

Final Answer:

The value of ab + bc + ca is 11.