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

Concept

To compare two variables that are each defined by a separate quadratic equation, solve each equation on its own to get its roots, then compare every root of the first variable against every root of the second. A single overall relation (≥, ≤, >, < or =) holds only when it is true for all such pairings; if some pairs give one sign and others give equality or the opposite sign, the relation must be widened or declared indeterminate. A perfect-square quadratic gives one repeated root.

Application

1. Equation I: 4x2 + 20x + 25 = 0 is a perfect square, since it equals (2x + 5)2 = 0. So 2x + 5 = 0, giving the single repeated root x = −5/2 = −2.5.

2. Equation II: 2y2 + 11y + 15 = 0. Factor by splitting the middle term: 2y2 + 6y + 5y + 15 = 2y(y + 3) + 5(y + 3) = (2y + 5)(y + 3) = 0. So y = −5/2 = −2.5 or y = −3.

3. Compare x = −2.5 with each y value: against y = −2.5 we get x = y; against y = −3 we get −2.5 > −3, i.e. x > y.

4. Across both cases x is never less than y — it is either equal to y or greater than y — so the relation that holds for every pair is x ≥ y.

Cross-check

Place the values on a number line: x sits at −2.5, while y is at −2.5 or −3. The smallest y (−3) lies to the left of x, and the largest y (−2.5) coincides with x. There is no y to the right of x, so x can never be strictly less than y; the tightest single relation covering both equal and greater cases is x ≥ y.