Let p and q be the roots of x^2 + p x + q = 0. Determine the correct values of (p, q) consistent with this definition.
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Ap = 1 and q = -2
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Bp = 0 and q = 1
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Cp = -2 and q = 0
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Dp = -2 and q = 1
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Ep = 2 and q = -1
Answer
Correct Answer: p = 1 and q = -2
Explanation
Introduction / Context: Here the symbolic coefficients p and q also equal the roots of their own quadratic. This creates equations among p and q via Vieta’s relations that must be satisfied simultaneously.
Given Data / Assumptions:
- Equation: x^2 + px + q = 0.
- Roots are p and q themselves.
- Real values expected.
Concept / Approach: For monic x^2 + px + q = 0 with roots r1 = p and r2 = q: sum r1 + r2 = −p and product r1*r2 = q (Vieta). This yields a system in p and q.
Step-by-Step Solution:
Sum: p + q = −p → q = −2p.Product: pq = q.Either q = 0 or p = 1.If q = 0 then q = −2p → p = 0 (gives (0, 0), not in options). If p = 1 then q = −2.Therefore (p, q) = (1, −2).Verification / Alternative check: With p = 1, q = −2, the quadratic x^2 + x − 2 = 0 has roots 1 and −2, matching p and q.
Why Other Options Are Wrong: They violate either the sum or product condition; check Vieta’s relations to see inconsistencies.
Common Pitfalls: Missing the special solution q = 0, p = 0 (not listed) or confusing coefficient p with root p without applying Vieta correctly.
Final Answer: p = 1 and q = −2