Find the roots exactly: Solve 2x^2 − 11x + 15 = 0.

Introduction / Context:
Factoring is often the fastest way to solve a quadratic with integer coefficients if a suitable factor pair exists. Here the numbers factor cleanly, yielding rational roots without the quadratic formula.

Given Data / Assumptions:

• Equation: 2x^2 − 11x + 15 = 0
• Look for factorization into (2x − m)(x − n) with mn = 15 and 2n + m = 11.

Concept / Approach:
We seek integers m, n such that 2x^2 − 11x + 15 = (2x − 5)(x − 3). Then set each factor to zero to get the roots. This is quicker than the quadratic formula and equally valid.

Step-by-Step Solution:

Verification / Alternative check:
Expand (2x − 5)(x − 3) = 2x^2 − 6x − 5x + 15 = 2x^2 − 11x + 15, confirming correctness.

Why Other Options Are Wrong:

• Other sign variants do not satisfy the original equation when substituted.

Common Pitfalls:
Misassigning signs to factor pairs of 15, or mixing the coefficient 2 into the wrong factor, leading to cross-term errors.

Final Answer:
3 and 5/2