A root of the quadratic equation ax^2 + bx + c = 0, where a, b and c are rational numbers, is 5 + 3√3. What is the value of (a^2 + b^2 + c^2) / (a + b + c)?

Introduction / Context:
Quadratic equations with irrational roots often appear in algebra and aptitude tests, especially when the coefficients are restricted to rational numbers. If a quadratic equation with rational coefficients has a root that involves a square root of a non perfect square, then its conjugate must also be a root. This question gives one such root, 5 + 3√3, and asks you to compute a ratio involving the coefficients a, b and c of the quadratic. It tests understanding of conjugate roots and manipulation of symmetric expressions in the coefficients.

Given Data / Assumptions:

• A root of ax^2 + bx + c = 0 is 5 + 3√3.
• a, b and c are rational numbers.
• The other root must therefore be 5 - 3√3.
• We must evaluate (a^2 + b^2 + c^2) / (a + b + c).

Concept / Approach:
Because the coefficients are rational and one root is 5 + 3√3, the other root is 5 - 3√3. The monic quadratic with these roots is (x - (5 + 3√3))(x - (5 - 3√3)) = x^2 - 10x - 2. Any quadratic with these roots and rational coefficients is a constant multiple of this polynomial, say a(x^2 - 10x - 2). That means we can safely take a = 1, b = -10 and c = -2 without changing the value of any symmetric ratio where the same scalar is present in all coefficients. Then we compute (a^2 + b^2 + c^2) / (a + b + c) for this convenient set of coefficients.

Step-by-Step Solution:
Step 1: Write the quadratic with roots 5 + 3√3 and 5 - 3√3 as (x - (5 + 3√3))(x - (5 - 3√3)). Step 2: Expand: (x - 5 - 3√3)(x - 5 + 3√3) = (x - 5)^2 - (3√3)^2. Step 3: Compute (x - 5)^2 = x^2 - 10x + 25 and (3√3)^2 = 9 × 3 = 27. Step 4: Subtract to get x^2 - 10x + 25 - 27 = x^2 - 10x - 2. Step 5: Thus, one possible quadratic is x^2 - 10x - 2 = 0, corresponding to a = 1, b = -10, c = -2. Step 6: Compute a^2 + b^2 + c^2 = 1^2 + (-10)^2 + (-2)^2 = 1 + 100 + 4 = 105. Step 7: Compute a + b + c = 1 + (-10) + (-2) = 1 - 10 - 2 = -11. Step 8: Evaluate the ratio: (a^2 + b^2 + c^2) / (a + b + c) = 105 / (-11) = -105/11.

Verification / Alternative check:
If the actual quadratic were k(x^2 - 10x - 2) = 0 for some nonzero rational k, then a = k, b = -10k and c = -2k. In that case a^2 + b^2 + c^2 equals k^2(1 + 100 + 4) = 105k^2, and a + b + c equals k(1 - 10 - 2) = -11k. The ratio becomes 105k^2 / (-11k) = -105k/11. However, since k is nonzero, we can choose k = 1 to simplify calculations without changing the structure of the problem, and the resulting ratio -105/11 is consistent. This confirms that the value is independent of the particular scalar multiple of the polynomial chosen.

Why Other Options Are Wrong:

• Options a (35/3) and b (37/3) are unrelated to the symmetric expression derived from the correct coefficients and likely come from incorrect expansions or mistaken root sums.
• Option d (-105/13) uses the right numerator but the wrong denominator, indicating a miscalculation of a + b + c.
• Option e (14) completely ignores the negative sign and the fraction structure.

Common Pitfalls:
Students sometimes forget that with rational coefficients, complex or irrational roots must come in conjugate pairs. Others might try to find a, b and c directly from one root without using the conjugate, leading to messy algebra. Another frequent error is expanding the product of factors with surds incorrectly, especially when handling the middle term or the difference of squares. Writing each step clearly and recalling that (x - r1)(x - r2) expands to x^2 - (r1 + r2)x + r1 r2 keeps the process under control.

Final Answer:
The value of (a^2 + b^2 + c^2) / (a + b + c) is -105/11.