Sum of roots equals sum of squares: For x^2 + px + q = 0, if (sum of roots) = (sum of squares of roots), find the relation between p and q.
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Ap^2 − q^2 = 0
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Bp^2 + q^2 = 2q
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Cp^2 + p = 2q
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Dq^2 + q^2 = 2p
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Ep^2 − p = 2q
Answer
Correct Answer: p^2 + p = 2q
Explanation
Introduction / Context:Using Vieta’s formulas, we can relate the sum and product of roots of a quadratic to its coefficients. The condition equating the sum of the roots to the sum of their squares produces an algebraic relation between p and q, which we simplify to the requested form.
Given Data / Assumptions:
- Equation: x^2 + px + q = 0
- Sum of roots S = −p
- Product P = q
- Condition: S = (sum of squares) = S^2 − 2P
Concept / Approach:Compute the sum of squares via S^2 − 2P, set equal to S, and rearrange to isolate p, q in a neat relation. This is a classic symmetric manipulation avoiding any need to find individual roots.
Step-by-Step Solution:
S = S^2 − 2P ⇒ −p = p^2 − 2qRearrange: p^2 + p − 2q = 0 ⇒ p^2 + p = 2qVerification / Alternative check:Choose a numeric pair (p, q) satisfying p^2 + p = 2q, generate the quadratic, compute its roots, and verify that r1 + r2 equals r1^2 + r2^2. The identity is exact.
Why Other Options Are Wrong:
- Other relations do not follow from S = S^2 − 2P and conflict with Vieta’s formulas.
Common Pitfalls:Sign errors when substituting S = −p; remember P = q. Do not confuse S^2 − 2P with (r1 − r2)^2.
Final Answer:p^2 + p = 2q