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Quadratic Equation Questions for Bank Exams: Roots, Signs and Comparing x and y

Factorise by sum and product • Sign rule for roots • Leading coefficient not 1 • The five answer options • x² = k versus √k

Banking 📅 Last updated: 2026-10-23

Bank prelims papers regularly include a set of questions that each give two quadratic equations, one in x and one in y. You solve both and pick the relation between x and y from five options:

  • x > y
  • x < y
  • x ≥ y
  • x ≤ y
  • x = y, or the relationship cannot be established

Each equation usually has two roots, so the relation must hold for every pairing of an x root with a y root. If it holds for some pairs and fails for others, the answer is "cannot be established". Speed comes from factorising quickly and reading the signs of the roots before doing any arithmetic.

For x² + bx + c = 0, find two numbers that multiply to c and add to b. If those numbers are p and q, the equation is (x + p)(x + q) = 0, and the roots are −p and −q.

Worked example 1: x² − 7x + 12 = 0 and y² − 11y + 30 = 0

For x: −3 and −4 multiply to 12 and add to −7, so x = 3 or 4.

For y: −5 and −6 give y = 5 or 6. Every x is smaller than every y, so x < y.

The sum of the roots is −b ÷ a and the product is c ÷ a. These two facts let you check roots instantly without substituting back.

The signs of b and c in x² + bx + c = 0 tell you the signs of the roots before you find them.

Sign of bSign of cRoots
++both negative
−+both positive
+ or −−one positive, one negative

If x's roots are both positive and y's are both negative, the answer is x > y without any calculation. If either equation has roots of opposite signs, the answer is often "cannot be established", but check: the other equation's roots may still sit entirely above or below both.

Worked example 2: both negative

x² + 9x + 20 = 0 gives x = −5 or −4. y² + 7y + 12 = 0 gives y = −4 or −3. The largest x equals the smallest y, so x ≤ y.

Worked example 3: opposite signs

x² − x − 6 = 0 gives x = 3 or −2. y² − 5y + 6 = 0 gives y = 2 or 3. x = 3 is at least every y, but x = −2 is below both, so the relationship cannot be established.

For ax² + bx + c = 0, multiply a by c, find two numbers with that product and sum b, and split the middle term. Then factorise in pairs.

Worked example 4: 2x² − 11x + 15 = 0

a × c = 30. The numbers −5 and −6 multiply to 30 and add to −11. So 2x² − 5x − 6x + 15 = x(2x − 5) − 3(2x − 5) = (x − 3)(2x − 5), giving x = 3 or 2.5.

Worked example 5: 3x² − 17x + 20 = 0

a × c = 60, and −5 and −12 add to −17. The equation becomes (x − 4)(3x − 5) = 0, so x = 4 or 5/3.

A frequent twist gives one equation as x² = 49 and the other as y = √49. The first has two roots, +7 and −7. The second uses the square root symbol, which means only the positive root, 7.

Worked example 6: x² = 49 and y = √49

x = 7 or −7, and y = 7. x is never greater than y and equals it when x = 7, so x ≤ y.

  • ✘ Comparing only the larger roots instead of every pairing.
  • ✘ Choosing "x > y" when one pair is equal. That case is "x ≥ y".
  • ✘ Giving √k two values. The √ symbol means the positive root only.
  • ✘ Forgetting to flip signs: (x − 4) = 0 gives x = +4.

Quadratics also appear disguised in x + 1/x problems, and the expansions used here are on the algebraic identities page. More shortcuts are in the Quant cheat code.

Q1. What is the larger root of x² − 9x + 20 = 0? easy

✅ Answer: (b) 5

💡 Explanation: Find two numbers that add to 9 and multiply to 20: 4 and 5. The equation is (x − 4)(x − 5) = 0, so the larger root is 5.

Q2. What is the sum of the roots of 2x² − 7x + 3 = 0? easy

✅ Answer: (a) 3.5

💡 Explanation: For ax² + bx + c = 0 the sum of the roots is −b ÷ a = 7 ÷ 2 = 3.5. The roots themselves are 3 and 1/2.

Q3. I. x² − 8x + 15 = 0 II. y² − 3y + 2 = 0. Which relation holds? easy

✅ Answer: (c) x > y

💡 Explanation: x = 3 or 5, and y = 1 or 2. Every value of x is larger than every value of y, so x > y.

Q4. I. x² + 11x + 30 = 0 II. y² + 13y + 42 = 0. Which relation holds? medium

✅ Answer: (d) x ≥ y

💡 Explanation: x = −5 or −6, and y = −6 or −7. Each x is greater than or equal to each y, with x = y = −6 possible, so x ≥ y.

Q5. I. x² − 2x − 15 = 0 II. y² + 5y + 6 = 0. Which relation holds? medium

✅ Answer: (b) Relationship cannot be established

💡 Explanation: x = 5 or −3, and y = −2 or −3. x = 5 is above both values of y, but x = −3 is below y = −2. No single relation holds for every pair.

Q6. What is the positive root of x² − x − 12 = 0? easy

✅ Answer: (a) 4

💡 Explanation: Two numbers that multiply to −12 and add to −1 are −4 and +3, so (x − 4)(x + 3) = 0. The roots are 4 and −3, and the positive one is 4.

When is the answer 'x = y'?

Only when both equations have the same single value, for example when each is a perfect square with the same repeated root. If x can be two values and y two values, x = y for every pair is impossible, so check for ≥ or ≤ instead.

Should I always use the quadratic formula?

No. In bank exams the roots are almost always whole numbers or simple fractions, so factorising by sum and product is faster. Keep the formula as a fallback for the rare equation that will not factorise neatly.

All problems and solutions on this page are original. No outside facts are relied on; every numeric answer is recomputed by software before publishing.

How this page was prepared. Drafted with AI assistance (Claude (Anthropic AI model)) from the sources listed above. 10 calculations on this page were recomputed by software and matched. Automated checks also covered structure, internal links and duplicate text. Software checks cannot confirm facts. Reviewed by praveenkumar on 2026-10-23 against the sources listed above.

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