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.

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:

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