Evaluate a symmetric expression in roots: If a and b are roots of x^2 − 6x + 6 = 0, find the value of 2(a^2 + b^2).

Introduction / Context:This is a symmetric expression in the roots of a quadratic. Rather than solving for the roots explicitly, we use Vieta’s formulas together with the identity a^2 + b^2 = (a + b)^2 − 2ab to evaluate the expression in a few steps.

Given Data / Assumptions:

• x^2 − 6x + 6 = 0 ⇒ a + b = 6, ab = 6
• Compute 2(a^2 + b^2)

Concept / Approach:Calculate a^2 + b^2 using the sum and product, then multiply by 2 at the end. This avoids any approximate decimal roots and keeps the computation exact and quick.

Step-by-Step Solution:

Verification / Alternative check:If desired, approximate roots via formula and square individually, but the identity is exact and simpler, leading to the same result 48.

Why Other Options Are Wrong:

• 40, 42, 46, 36: These arise from arithmetic slips such as forgetting the factor 2 or miscomputing 2ab.

Common Pitfalls:Mixing up ab with a^2b^2, or using (a − b)^2 instead of (a + b)^2 − 2ab. Stick to the standard identity for sum of squares.

Final Answer:48