Condition for exactly one real (repeated) root of x^2 − p x + q = 0 (with p, q ∈ ℝ): State the correct discriminant condition ensuring a single real root.

Introduction / Context:
A quadratic equation ax^2 + bx + c = 0 has its number and nature of real roots governed by the discriminant D = b^2 − 4ac. Here a = 1, b = −p, c = q. The phrase “exactly one real root” means a repeated real root (a double root), which occurs precisely when the discriminant equals zero.

Given Data / Assumptions:

• Equation: x^2 − p x + q = 0
• Parameters: p, q ∈ ℝ, with leading coefficient 1 ≠ 0
• We seek the condition for one real (repeated) root.

Concept / Approach:
Use the discriminant test. For a quadratic ax^2 + bx + c = 0: D > 0 ⇒ two distinct real roots; D = 0 ⇒ one real repeated root; D < 0 ⇒ no real roots (complex conjugate pair). Identify b and c in terms of p and q, compute D, and enforce D = 0.

Step-by-Step Solution:
Here a = 1, b = −p, c = qDiscriminant D = b^2 − 4ac = (−p)^2 − 4*1*q = p^2 − 4qExactly one real root ⇔ D = 0 ⇔ p^2 − 4q = 0Therefore the required condition is p^2 = 4q

Verification / Alternative check:
When p^2 = 4q, the quadratic becomes (x − p/2)^2 = 0 after completing the square, which clearly has a single repeated root at x = p/2.

Why Other Options Are Wrong:
p^2 < 4q gives D < 0 (no real roots). p^2 > 4q gives D > 0 (two distinct real roots). “No condition” and “p^2 ≤ 4q” do not specifically guarantee exactly one real root.

Common Pitfalls:
Confusing “at least one real root” (D ≥ 0) with “exactly one real root” (D = 0). The equality case is essential here.

Final Answer:
p^2 = 4q