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

Concept: For two quadratic equations in x and y, first find all roots of each equation. A single inequality (x > y, x < y, x ≥ y or x ≤ y) can be declared the answer ONLY if that same relation holds for EVERY pairing of an x-root with a y-root. If different pairings give different directions, no single relation is valid and the answer is “no relation can be established”.

Application — Equation I: 3x2 − 14x + 15 = 0.

1. Factorise: 3x2 − 14x + 15 = (3x − 5)(x − 3) = 0.

2. So x = 5/3 ≈ 1.67 or x = 3.

Application — Equation II: 6y2 − 31y + 18 = 0.

1. Factorise: 6y2 − 31y + 18 = (3y − 2)(2y − 9) = 0.

2. So y = 2/3 ≈ 0.67 or y = 9/2 = 4.5.

Cross-check — compare every x with every y:

Some pairings give x < y and others give x > y, so the direction is not consistent. Therefore no single relation holds between x and y — the value “x = y or no relation can be established” is the result.