Let x be the greater real root of x^2 − x − 12 = 0 and y be the greater real root of y^2 + 5y + 6 = 0. Compare x and y.

Introduction / Context: Straightforward factoring reveals both sets of roots, and the greater ones are compared directly as per the clarified convention.

Given Data / Assumptions:

• x^2 − x − 12 = 0 ⇒ (x − 4)(x + 3) = 0 ⇒ greater x = 4.
• y^2 + 5y + 6 = 0 ⇒ (y + 2)(y + 3) = 0 ⇒ greater y = −2.

Concept / Approach: Factor both quadratics. Identify greater real roots. Compare numerically.

Step-by-Step Solution:

Verification / Alternative check: Plugging back confirms each value solves the respective equation.

Why Other Options Are Wrong: They contradict 4 > −2.

Common Pitfalls: Picking the smaller root for either quadratic or sign mistakes during factoring.

Final Answer: If x > y