Exactly one real root (repeated) condition: Determine all real values of p such that x^2 + 5px + 16 = 0 has exactly one real root (i.e., equal roots).

Introduction / Context:
“Exactly one real root” for a quadratic means the graph touches the x-axis at a single point: the discriminant must be zero. We find all p that make Δ = 0 for x^2 + 5px + 16 = 0.

Given Data / Assumptions:

• a = 1, b = 5p, c = 16.
• p is real.

Concept / Approach:
Set Δ = b^2 − 4ac = 0 and solve for p. This yields the exact parameter values where the quadratic has a repeated real root.

Step-by-Step Solution:

Verification / Alternative check:
For p = 8/5, the quadratic is x^2 + 8x + 16 = 0 ⇒ (x + 4)^2 = 0. For p = −8/5, x^2 − 8x + 16 = 0 ⇒ (x − 4)^2 = 0. Both have a single (repeated) real root.

Why Other Options Are Wrong:

• −8/5 < p < 8/5: Gives Δ < 0 (no real roots).
• p ≤ −8/5 or p ≥ 8/5: Includes Δ > 0 as well as Δ = 0; not “exactly one.”
• No real value of p: False; two real values exist.

Common Pitfalls:
Confusing “at least one real root” (Δ ≥ 0) with “exactly one real root” (Δ = 0). Here the equality is essential.

Final Answer: