Comparing Two Variables: The x and y Format

The heart of the SBI Clerk quadratic block is the comparison itself. You solve two equations, one in x and one in y, then choose the correct relation from a fixed list of five options. Getting the mechanics of this comparison right is what converts your roots into the actual mark, so it deserves careful, deliberate practice.

The Five Standard Options

Almost every bank comparison offers the same five relations, so learn them as a set: x > y, x < y, x ≥ y, x ≤ y, and 'x = y or relation cannot be established'. The ≥ and ≤ options appear when one root of x equals a root of y, and the 'cannot be established' option appears when the roots overlap or interleave.

The Correct Method: Compare Every Root

You must compare each root of x against each root of y, not just the sums or the first values you find. Write the two x-roots and the two y-roots on a number line in your head. If both x-roots sit to the right of both y-roots, then x > y; if they interleave — one x-root larger, one smaller than the y-roots — the relation cannot be established, which is the option most candidates miss.

x > y only if BOTH x-roots ≥ both y-roots AND at least one is strictly greater

When the Answer Is 'Cannot Be Established'

This option trips up more candidates than any other. It is correct whenever the two ranges of roots overlap, so that x is sometimes larger and sometimes smaller than y depending on which roots you pick. Always check for interleaving before selecting a strict inequality; the examiner includes this trap precisely because rushed candidates compare only one pair of roots.

Solved Example

Example: x² − 7x + 12 = 0 gives x = 3, 4. y² − 9y + 20 = 0 gives y = 4, 5. Compare: x-roots 3, 4 and y-roots 4, 5. Since 3 < 5 and 4 ≤ 4, x never exceeds y and sometimes is smaller, so the answer is x ≤ y.

✗ Comparing only the larger roots 4 and 5 to conclude x < y  |  ✓ Comparing all four roots to see x ≤ y (because 4 = 4)

Placing Roots on a Number Line

The safest way to compare is to picture all four roots on a single number line. Mark the two x-roots and the two y-roots in order of size, then look at how the two pairs sit relative to each other. If the entire x-pair lies to the right of the entire y-pair, x is greater; if it lies entirely to the left, x is smaller; if they touch at one shared value, you get the ≥ or ≤ option; and if they overlap, no fixed relation holds. This picture removes almost all comparison errors because it forces you to consider every root at once rather than the one that happens to catch your eye.

A Second Worked Comparison

Consider x² − 5x + 6 = 0 giving x = 2, 3 and y² − 8y + 15 = 0 giving y = 3, 5. On the number line the x-roots 2 and 3 both sit at or below the y-roots 3 and 5, and since 3 = 3 the relation is not strict, so the answer is x ≤ y. Had the y-roots been 4 and 5 instead, both x-roots would be strictly smaller and the answer would sharpen to x < y. Small shifts in the roots change the option, which is exactly why you must compare the full sets every time.