Equations: I. a^2 - 7a + 12 = 0 II. b^2 - 3b + 2 = 0 Solve both equations to find the possible values of a and b, and determine the relationship between a and b.

Introduction / Context:
This question gives you two quadratic equations in a and b and asks you to determine the relationship between a and b. Instead of asking for numerical values directly, it focuses on comparing all possible roots of each equation. This is a common pattern in comparison questions where you must consider all solutions and then see whether a consistent relation holds in every case.

Given Data / Assumptions:

Equation I: a^2 - 7a + 12 = 0.
Equation II: b^2 - 3b + 2 = 0.
We need to solve each equation and compare all possible values of a and b.

Concept / Approach:
A quadratic equation of the form x^2 + px + q = 0 can be factored or solved using standard methods. After finding the roots for a and b, we check every possible pairing of (a, b) and see whether a > b, a < b, a = b, or if the comparison varies depending on the pair. If the same inequality holds for every valid combination, we can assert that relationship confidently.

Step-by-Step Solution:
Solve Equation I: a^2 - 7a + 12 = 0. Factor the quadratic: 12 factors as 3 × 4 and 3 + 4 = 7, so a^2 - 7a + 12 = (a - 3)(a - 4) = 0. Thus, a has two possible values: a = 3 or a = 4. Solve Equation II: b^2 - 3b + 2 = 0. Factor the quadratic: 2 factors as 1 × 2 and 1 + 2 = 3, so b^2 - 3b + 2 = (b - 1)(b - 2) = 0. Thus, b has two possible values: b = 1 or b = 2. Now compare all possible pairs: If a = 3 and b = 1 → a > b. If a = 3 and b = 2 → a > b. If a = 4 and b = 1 → a > b. If a = 4 and b = 2 → a > b. In every possible combination of roots, a is strictly greater than b.

Verification / Alternative check:
You can visualise the roots on the number line: the set of possible a values is {3, 4} and the set of possible b values is {1, 2}. Since every element in {3, 4} is greater than every element in {1, 2}, the relationship a > b always holds. There is no case where a equals or is less than b, confirming the comparison.

Why Other Options Are Wrong:
Options suggesting a < b or a ≤ b are directly contradicted by the values found. The option saying the relationship cannot be established would be correct only if different pairs gave different inequalities, but here the inequality is consistent across all pairs. Similarly, a ≥ b is weaker than necessary; we can say more precisely that a is strictly greater than b in all cases.

Common Pitfalls:
One common mistake is to solve for only one root of each quadratic and ignore the other possibilities. Another is to assume that because there are multiple roots, no single relationship can be established; in reality, if every combination yields the same inequality, the relationship is perfectly well-defined.

Final Answer:
In all possible cases, a is greater than b. Therefore, the correct choice is if a > b.