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

Concept

In these paired-equation problems, solve each quadratic to get all roots of x and all roots of y, then compare every x-root with every y-root. The keyed relation follows the standard convention: choose a strict relation (x > y or x < y) when one variable's roots are all strictly larger (or all strictly smaller) than the other's; choose an 'or equal' relation (x ≥ y or x ≤ y) only when the roots touch at equality in at least one pair while never crossing; and choose 'no relation' only when the comparison flips direction across pairs.

Application — Equation I (in x)

1. 4x2 - 16x + 16 = 0; divide every term by 4 to get x2 - 4x + 4 = 0.

2. This is a perfect square: (x - 2)2 = 0.

3. So x = 2 is a repeated root — the only value of x is 2.

Application — Equation II (in y)

1. 10y2 - 29y + 21 = 0; split the middle term using 10 × 21 = 210 = 14 × 15 and 14 + 15 = 29.

2. 10y2 - 14y - 15y + 21 = 0 → 2y(5y - 7) - 3(5y - 7) = 0.

3. (2y - 3)(5y - 7) = 0, so y = 3/2 = 1.5 or y = 7/5 = 1.4.

Cross-check — compare the values

The single x-value is 2; the two y-values are 1.5 and 1.4, both below 2.

• 2 > 1.5 ✓

• 2 > 1.4 ✓

Every root of one variable lies strictly on the same side of every root of the other, with no pair ever equal and no flip in direction. By the convention above this is a strict relation, and the larger variable is x. (Quick check: putting one solved y-root back into equation II returns 0, confirming the roots.)