If a(a + b + c) = 45, b(a + b + c) = 75 and c(a + b + c) = 105 for real numbers a, b and c, then what is the value of the sum a^2 + b^2 + c^2?

Introduction / Context:
This algebra problem tests how to use a common factor (a + b + c) to express several related products and then deduce the individual values of a, b, and c. Once those values are known, computing a^2 + b^2 + c^2 is straightforward. Questions of this type encourage thinking in terms of shared structure and factoring.

Given Data / Assumptions:

a(a + b + c) = 45.
b(a + b + c) = 75.
c(a + b + c) = 105.
a, b, and c are real numbers.
We need a^2 + b^2 + c^2.

Concept / Approach:
We notice that each equation has the common factor S = a + b + c. Writing aS = 45, bS = 75, and cS = 105 allows us to express a, b, and c in terms of S. Then we can use the fact that a + b + c equals S itself to form a single equation in S and solve it. Once S is known, we find a, b, and c individually and compute their squares and sum.

Step-by-Step Solution:
Let S = a + b + c.From a(a + b + c) = 45 we get aS = 45, so a = 45 / S.From b(a + b + c) = 75 we get bS = 75, so b = 75 / S.From c(a + b + c) = 105 we get cS = 105, so c = 105 / S.Now sum a + b + c and set equal to S: (45 / S) + (75 / S) + (105 / S) = S.This gives (45 + 75 + 105) / S = S.Compute 45 + 75 + 105 = 225, so 225 / S = S.Therefore S^2 = 225, so S = 15 or S = −15.From the original equations, a, b, and c are 45 / S, 75 / S, and 105 / S. Taking S = 15 gives a = 3, b = 5, c = 7, which are simple positive values.If S = −15, then a, b, and c are negative but their squares will be the same as in the positive case, so a^2 + b^2 + c^2 will be unchanged.Compute a^2 + b^2 + c^2 = 3^2 + 5^2 + 7^2 = 9 + 25 + 49 = 83.

Verification / Alternative check:
Substitute a = 3, b = 5, c = 7, and S = 15 into the original equations. We get aS = 3 * 15 = 45, bS = 5 * 15 = 75, and cS = 7 * 15 = 105, which matches the given conditions. This confirms that our values are consistent and that a^2 + b^2 + c^2 really is 83.

Why Other Options Are Wrong:
Values such as 75, 217, 225, and 133 would correspond to different combinations of a, b, and c. However, those combinations would not satisfy all three proportional equations simultaneously. Only the combination that leads to 3, 5, and 7 (or their negatives) fits the given products and therefore only 83 is valid for the sum of squares.

Common Pitfalls:
Some students attempt to eliminate variables pairwise without introducing the sum S, which quickly becomes messy. Others forget that a + b + c equals S and treat the three equations as independent. Introducing S explicitly and solving for it keeps the algebra organized and simple.

Final Answer:
The value of a^2 + b^2 + c^2 is 83.