Quadratic equations for IBPS Clerk

In this format you get two equations, usually one in x and one in y. You solve both and decide how x and y compare. The algebra is Class 10 level. The marks are lost in two places: slow factorising, and comparing only some of the roots instead of all of them.

Quadratic comparison has appeared in clerk papers as a small block of questions, more often in the mains than the prelims. Each question can be done in about 30 seconds with practice, which makes it a good block to master even if it carries only a few marks.

The answer options

The options are nearly always these five:

1. x > y
2. x < y
3. x ≥ y
4. x ≤ y
5. x = y, or no relation can be established

Your task is to find which statement is true for every pair of an x-root with a y-root.

Step 1: factorise by splitting the middle term

For x² + bx + c = 0, find two numbers that multiply to c and add to b.

• x² − 11x + 30 = 0: numbers −5 and −6 (product 30, sum −11). So (x − 5)(x − 6) = 0 and x = 5 or 6.

For ax² + bx + c = 0, find two numbers that multiply to a × c and add to b.

• 2x² − 9x + 10 = 0: a × c = 20. Numbers −4 and −5. Split: 2x² − 4x − 5x + 10 = 2x(x − 2) − 5(x − 2) = (2x − 5)(x − 2). So x = 2 or 5/2.

The fast way to read the roots

Once you have the two numbers p and q, the roots are −p ÷ a and −q ÷ a. Flip the signs and divide by a. For 2x² − 9x + 10 = 0, the numbers are −4 and −5, so the roots are 4/2 = 2 and 5/2 = 2.5. No need to write the factors.

Why it works: if p + q = b and pq = ac, then ax² + bx + c = (1/a)(ax + p)(ax + q). Setting each bracket to zero gives x = −p/a and x = −q/a. You can check it by expanding: (ax + p)(ax + q) = a²x² + a(p + q)x + pq = a(ax² + bx + c).

The sign rule

The signs of b and c tell you the signs of the roots before you factorise. This is useful for spotting "no relation" answers early.

Why it works: for ax² + bx + c = 0 with a positive, the sum of the roots is −b/a and the product is c/a. A positive product means both roots have the same sign, and the sum then tells you which sign. A negative product means the roots have opposite signs.

Step 2: compare every pair

List both x-roots and both y-roots, then check each of the four pairings. A number line helps.

A quick test: compare the smallest x with the largest y, and the largest x with the smallest y. If smallest x ≥ largest y, then x ≥ y (strictly greater if no equality). If largest x ≤ smallest y, then x ≤ y. Otherwise there is no relation.

Worked examples

Example 1. I. x² − 11x + 30 = 0 · II. y² − 7y + 12 = 0

1. x: numbers −5, −6, so x = 5, 6.
2. y: numbers −3, −4, so y = 3, 4.
3. Smallest x (5) is greater than largest y (4). x > y.

Example 2. I. 2x² + 11x + 15 = 0 · II. y² + 5y + 6 = 0

1. x: a × c = 30, numbers 5 and 6. Roots −5/2 and −6/2, so x = −2.5, −3.
2. y: numbers 2 and 3, so y = −2, −3.
3. Pairs: x = −2.5 is less than y = −2 but greater than y = −3. No relation can be established.

This is the classic trap. The roots look close and many candidates mark x ≤ y without checking −2.5 against −3.

Example 3. I. 3x² − 10x + 8 = 0 · II. y² − 5y + 6 = 0

1. x: a × c = 24, numbers −4 and −6. Roots 4/3 and 6/3, so x = 1.33, 2.
2. y: numbers −2, −3, so y = 2, 3.
3. Largest x (2) equals smallest y (2). Every other pair has x smaller. x ≤ y.

Example 4. I. x² = 81 · II. y² + 21y + 110 = 0

1. x² = 81 gives x = 9 or −9. Never drop the negative root.
2. y: numbers 10 and 11, so y = −10, −11.
3. Even the smaller x (−9) is greater than the larger y (−10). x > y.

A ± root does not automatically mean "no relation". Check it.

Example 5. I. 6x² − 5x + 1 = 0 · II. 12y² − 7y + 1 = 0

1. x: a × c = 6, numbers −2 and −3. Roots 2/6 and 3/6, so x = 1/3, 1/2.
2. y: a × c = 12, numbers −3 and −4. Roots 3/12 and 4/12, so y = 1/4, 1/3.
3. Smallest x (1/3) equals largest y (1/3). x ≥ y.

Common mistakes

• Taking only the positive root of x² = 81 or y² = 49.
• Comparing only the "first" root of each equation.
• Forgetting to divide by a when a is not 1.
• Marking x > y when one pair is equal (that is x ≥ y).
• Copying a sign wrongly. Check with the sign rule: if b and c are both positive, both roots must be negative.

Practice

Aim for 30 seconds each.

1. I. x² − 13x + 42 = 0 · II. y² − 15y + 56 = 0
2. I. x² + 12x + 35 = 0 · II. y² + 7y + 12 = 0
3. I. 2x² − 9x + 10 = 0 · II. y² − 4y + 4 = 0
4. I. x² = 64 · II. y² − 17y + 72 = 0
5. I. 3x² + 11x + 10 = 0 · II. y² + 3y + 2 = 0
6. I. x² − x − 12 = 0 · II. y² + 7y + 12 = 0
7. I. 4x² − 12x + 9 = 0 · II. y² − 5y + 6 = 0

Answers:

1. x ≤ y. x = 6, 7; y = 7, 8. Largest x equals smallest y.
2. x < y. x = −5, −7; y = −3, −4. Largest x (−5) is below smallest y (−4).
3. x ≥ y. x = 2, 2.5; y = 2 (repeated). Smallest x equals y.
4. x ≤ y. x = 8, −8; y = 8, 9. Largest x equals smallest y.
5. No relation. x = −5/3, −2; y = −1, −2. x = −1.67 is above y = −2 but below y = −1.
6. x ≥ y. x² − x − 12 = (x − 4)(x + 3), so x = 4, −3; y = −3, −4. Smallest x equals largest y.
7. x < y. 4x² − 12x + 9 = (2x − 3)², so x = 1.5; y = 2, 3.

What to do next

• Factorise 20 quadratics a day for a week, writing only the roots.
• Practise the sign rule until you can predict root signs before solving.
• Use the smallest-x and largest-y test on every question until it is automatic.
• Pair this with inequalities in reasoning, which uses the same comparison logic, and see the numerical ability plan for where quadratics fit in your weeks.