Express α^2 + β^2 in terms of a, b, c: If α and β are roots of ax^2 + bx + c = 0 (a ≠ 0), find α^2 + β^2.

Introduction / Context:
This is a direct application of Vieta’s relations and algebraic identities. The sum and product of roots of a quadratic allow you to express symmetric functions like α^2 + β^2 without solving for the roots explicitly.

Given Data / Assumptions:

• α and β are roots of ax^2 + bx + c = 0.
• a ≠ 0 so that degree and relations are valid.

Concept / Approach:
Use the identity α^2 + β^2 = (α + β)^2 − 2αβ. From Vieta, α + β = −b/a and αβ = c/a. Substitute and simplify carefully to avoid sign or denominator errors.

Step-by-Step Solution:

Verification / Alternative check:
Pick a sample quadratic (e.g., x^2 − 5x + 6 = 0 with roots 2 and 3). Compute α^2 + β^2 = 4 + 9 = 13. Formula gives (25 − 12)/1 = 13. Checks out.

Why Other Options Are Wrong:
The ones with +2ac use the wrong sign; others have incorrect scaling or extraneous factors in the denominator.

Common Pitfalls:
Missing the minus sign in the identity and forgetting to square the −b/a properly.

Final Answer:
(b^2 − 2ac) / a^2