Quadratic Equations & Inequalities — IBPS PO Quant

The IBPS quadratic set is not "solve for x" — it's "here are two equations in x and y; what's the relation between x and y?" You solve both quadratics, get the two roots of each, and pick from a fixed key: x > y, x < y, x ≥ y, x ≤ y, or relation cannot be established. It's pure mechanics — factor by the sum–product method (the quadratic formula is rarely needed on IBPS's clean numbers), read the signs of the roots, and compare. At 0–5 marks these are fast, and the only real trap is knowing when the answer is "can't be determined".

1. What IBPS actually asks

Two equations, e.g. x² − 7x + 12 = 0 and y² − 9y + 20 = 0, and one question: the relationship between x and y. The answer key is fixed:

1. x > y 2. x < y 3. x ≥ y 4. x ≤ y 5. x = y or relation cannot be established.

You must compare every x-root with every y-root.

2. Factor by the sum–product method (skip the formula)

For x² + bx + c = 0, find two numbers that multiply to c and add to b; the roots are their negatives:

• x² − 7x + 12: two numbers multiplying to +12, adding to −7 → −3 and −4 ⇒ roots x = 3, 4.
• Sign rule: roots = −(those two numbers). If the middle term is −7x and constant +12, both factors are negative in the bracket (x−3)(x−4), so roots are +3, +4.

$x=3,4\text{ from }(x-3)(x-4)=0$

Reading signs fast (before factoring):

• c > 0: both roots have the same sign — same as the sign of −b (if −b positive, both positive).
• c < 0: roots have opposite signs.

This sign preview often tells you the answer before full factoring.

3. The comparison rule

Once you have x-roots {x₁, x₂} and y-roots {y₁, y₂}, compare each pair:

• If every x-value ≥ every y-value (with at least one strict) ⇒ x > y (or ≥ if equality possible).
• If the ranges overlap — some x bigger, some y bigger, or roots interleave ⇒ relationship cannot be established.
• Line up all four roots on a number line; if the x's and y's don't cleanly separate, the answer is "can't be determined".

The most common correct answer in these sets is "cannot be established" — don't force a > or < when the roots interleave.

4. The ≥ / ≤ subtlety

• x ≥ y (not x > y) is correct when the smallest x equals the largest y at one point but x is otherwise larger — i.e. equality is possible but x is never smaller.
• If all four roots happen to be equal or fully tie, it's x = y.
• A single shared value with the rest separating one way gives the ≥/≤ answer, not the strict one.

5. Quadratic inequalities (the newer variant)

Occasionally the equation is an inequality: solve x² − 5x + 6 < 0. Factor (x−2)(x−3) < 0 ⇒ the product is negative between the roots ⇒ 2 < x < 3. Rules:

• (x−a)(x−b) < 0 ⇒ x lies between a and b.
• (x−a)(x−b) > 0 ⇒ x lies outside [a, b].

Then compare the x-range with the y-range as before.

7. The protocol

1. Factor each quadratic by sum–product; use the sign preview (c>0 same sign, c<0 opposite) to sanity-check.
2. List all roots of x and of y.
3. Compare every x with every y — ideally on a number line.
4. Choose the key: clean separation ⇒ >, <, ≥, ≤; interleaving ⇒ cannot be established.
5. For inequalities, remember (x−a)(x−b) < 0 ⇒ between the roots; > 0 ⇒ outside.