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.

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:

Verification / 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