In the following quadratic comparison question, two equations are given: I. 2x^2 − 21x + 54 = 0 and II. y^2 − 14y + 49 = 0. After solving both equations, what is the correct relationship between x and y?

Introduction / Context:
This is another quadratic comparison problem. Here, two quadratic equations in variables x and y are given, and you must determine the relationship between any possible root of the first equation and any possible root of the second equation. These questions strengthen understanding of quadratic roots and inequality comparisons.

Given Data / Assumptions:

• Equation I: 2x^2 − 21x + 54 = 0.
• Equation II: y^2 − 14y + 49 = 0.
• x can be any root of Equation I, and y can be any root of Equation II.
• We look for a relationship that holds for all possible combinations.

Concept / Approach:
We first solve each quadratic to find its roots. Then we compare the numerical values of the roots. If every possible x is less than every possible y, then x < y. If the relationships vary, the conclusion would be that the relationship cannot be determined. Here, carefully comparing all roots is key.

Step-by-Step Solution:
Step 1: Factor Equation I: 2x^2 − 21x + 54 = 0. Step 2: Factorisation gives (x − 6)(2x − 9) = 0. Step 3: Roots for x are x = 6 and x = 9 / 2 = 4.5. Step 4: Factor Equation II: y^2 − 14y + 49 = 0. Step 5: This is a perfect square: (y − 7)^2 = 0. Step 6: So y has a single repeated root y = 7. Step 7: Now compare x and y for all combinations. Case 1: x = 6, y = 7. Then x < y. Case 2: x = 4.5, y = 7. Then x < y again. Step 8: In both possible cases, x is strictly less than y.

Verification / Alternative check:
We can check quickly: the smallest y value possible is 7, and the largest x value possible is 6 (since 6 > 4.5). Because even the maximum value of x is still less than the minimum value of y, we can confidently say that for every possible pair (x, y), x < y holds true.

Why Other Options Are Wrong:
x > y is impossible because both roots of x are less than 7.
x ≥ y and x ≤ y imply that x could equal y in at least one case, but there is no root of x equal to 7.
The combined statement “x = y or relation can’t be established between x and y” is incorrect because a clear consistent relationship does exist: x is always less.

Common Pitfalls:
Some students may only compare one root of x with y and incorrectly choose a non-committal option. Others may fail to recognise the perfect square form of Equation II and miscalculate the roots. Always determine all roots carefully and compare the extreme values to see if a consistent inequality exists.

Final Answer:
For all possible values, x < y.