In each of these questions, two equation (I) and (II) are given. You have to…

Concept: To compare the roots of two quadratics, factor each into the form (px − a)(qx − b) = 0 to read off its roots, list both root sets, then compare them pairwise. A single inequality (≤, <, >, ≥) holds only if it is true for every pairing of an x-root with a y-root; equality at any boundary forces the relation to use ≤ or ≥ rather than the strict form.

Application — solve Equation I (3x2 − 37x + 114 = 0):

1. Factor: 3x2 − 37x + 114 = (3x − 19)(x − 6) = 0.

2. Set each factor to zero: 3x − 19 = 0 gives x = 19/3 ≈ 6.33; x − 6 = 0 gives x = 6.

3. So the x-roots are 6 and 19/3 (≈ 6.33).

Application — solve Equation II (3y2 − 46y + 171 = 0):

1. Factor: 3y2 − 46y + 171 = (3y − 19)(y − 9) = 0.

2. Set each factor to zero: 3y − 19 = 0 gives y = 19/3 ≈ 6.33; y − 9 = 0 gives y = 9.

3. So the y-roots are 19/3 (≈ 6.33) and 9.

Compare the root sets pairwise:

• Smallest x = 6 against y = 19/3 and y = 9: 6 < 6.33 and 6 < 9, so x < y here.

• Largest x = 19/3 against y = 19/3: 6.33 = 6.33, so x = y at this boundary.

• Largest x = 19/3 against y = 9: 6.33 < 9, so x < y here.

In every pairing x is less than y, except at one boundary where x = y. No pairing makes x larger than y. Because equality occurs at the boundary, the strict relation x < y cannot be claimed for all cases; the correct, fully-true relation is x ≤ y.

Cross-check: the maximum x-value (19/3) equals the minimum y-value (19/3), and the minimum x-value (6) is below it — so the entire x-set lies at or below the entire y-set, confirming x ≤ y.