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.
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A(b^2 − 2ac) / (2a^2)
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B(b^2 + 2ac) / a^2
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C(b^2 + 2ac) / (a^2 a)
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D(b^2 − 2ac) / a^2
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E(b^2 − ac) / a^2
Answer
Correct Answer: (b^2 − 2ac) / a^2
Explanation
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:
α + β = −b/a ⇒ (α + β)^2 = b^2/a^2 αβ = c/a Therefore α^2 + β^2 = (α + β)^2 − 2αβ = (b^2/a^2) − 2(c/a) Put over a common denominator a^2: α^2 + β^2 = (b^2 − 2ac)/a^2Verification / 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