Equal roots condition: The quadratic x^2 + px + q = 0 has equal roots if and only if:

Introduction / Context:
A quadratic has equal (repeated) roots when its discriminant is zero. For x^2 + px + q = 0, the discriminant is Δ = p^2 − 4q. Setting Δ = 0 gives the precise condition connecting p and q.

Given Data / Assumptions:

• Quadratic: x^2 + px + q = 0.
• Real coefficients; equal roots desired.

Concept / Approach:
Use Δ = p^2 − 4q = 0 ⇒ p^2 = 4q. This is necessary and sufficient.

Step-by-Step Solution:

Verification / Alternative check:
If p^2 = 4q, then the quadratic is (x + p/2)^2 = 0, showing a double root at x = −p/2.

Why Other Options Are Wrong:

• p^2 = 2q or p^2 = −2q or p^2 = −4q: Do not make Δ = 0; they lead to unequal or complex roots depending on signs.

Common Pitfalls:
Confusing the sign in Δ or misplacing the 4. The standard discriminant for ax^2 + bx + c is b^2 − 4ac.

Final Answer: