Equal roots condition (find P): For 3x^2 − 5x + P = 0, determine the value(s) of P for which the equation has equal (repeated) real roots.

Introduction / Context:
A quadratic has equal real roots exactly when its discriminant is zero. We compute the discriminant for 3x^2 − 5x + P and set it to zero to find P.

Given Data / Assumptions:

• a = 3, b = −5, c = P.

Concept / Approach:
Δ = b^2 − 4ac = (−5)^2 − 4*3*P. Set Δ = 0 and solve for P.

Step-by-Step Solution:

Verification / Alternative check:
With P = 25/12 the quadratic becomes 3x^2 − 5x + 25/12 = 0. Completing the square confirms a repeated root.

Why Other Options Are Wrong:

• −25/12, 25/6, −25/6 give Δ ≠ 0, hence not equal roots.

Common Pitfalls:
Arithmetic slips with 4ac or sign errors when b is negative. Always compute Δ carefully.

Final Answer: