In each question two equations numbered (I) and (II) are given. You should…

Concept

To compare the roots of two quadratic equations, first find all roots of each equation by factorisation, then compare every value of x against every value of y. A fixed relation (>, <, ≥, ≤) holds only if it is true for all pairs of roots; if the value ranges overlap or the relation flips across pairs, no fixed relation can be established.

Application

1. Equation I: 2x2 + 19x + 42 = 0. Factorise: 2x2 + 12x + 7x + 42 = 0 → 2x(x + 6) + 7(x + 6) = 0 → (x + 6)(2x + 7) = 0.

2. So x = −6 or x = −7/2 = −3.5.

3. Equation II: 2y2 + 7y + 5 = 0. Factorise: 2y2 + 2y + 5y + 5 = 0 → 2y(y + 1) + 5(y + 1) = 0 → (y + 1)(2y + 5) = 0.

4. So y = −1 or y = −5/2 = −2.5.

5. Compare every pair: −6 < −2.5, −6 < −1, −3.5 < −2.5, and −3.5 < −1. In all four pairings the x-value is smaller.

Cross-check

The largest possible x is −3.5 and the smallest possible y is −2.5. Since the biggest x (−3.5) is still less than the smallest y (−2.5), the value ranges do not overlap, so x is strictly less than y for every combination. Hence the relation is x < y.