Quadratic equations for bank exams is a frequently asked topic in SBI PO, IBPS PO, IBPS Clerk and RBI Assistant. This guide covers quadratic equation tricks, the sign trick, comparison rules for x and y, and 20 solved quadratic equation questions from basic to advanced level. Scroll to the end to download the free practice set PDF.
What is a Quadratic Equation? Formulas for Bank Exams
An equation of the form ax² + bx + c = 0 (where a ≠ 0) is a quadratic equation. It has two roots.
Key formulas
- Sum of roots = −b/a
- Product of roots = c/a
- Discriminant D = b² − 4ac
- Roots by formula: x = [−b ± √D] / 2a
What D tells you
- D > 0: two distinct real roots
- D = 0: two equal roots
- D < 0: no real roots
Quadratic Equation Tricks: Factorization Method
Find two numbers whose product = a × c and sum = b. Split the middle term with them, then take common factors.
Example: x² − 7x + 12 = 0
- Product = 12, sum = −7, so the numbers are −3 and −4
- x² − 3x − 4x + 12 = x(x − 3) − 4(x − 3) = (x − 3)(x − 4)
- x = 3, 4
With a coefficient on x²: 2x² + 7x + 6 = 0
- a × c = 12, sum = 7, so the numbers are 3 and 4
- 2x² + 3x + 4x + 6 = x(2x + 3) + 2(2x + 3) = (2x + 3)(x + 2)
- x = −3/2, −2
Quadratic Formula Method (When Factorization Fails)
Use it when factorization is not visible within 10 seconds.
Example: 2x² − 6x + 3 = 0
- D = 36 − 24 = 12
- x = (6 ± √12) / 4 = (3 ± √3) / 2
Sign Trick for Quadratic Equations
For x² + bx + c = 0, the signs of b and c tell you the signs of the roots before you factorize.
| c | b | Roots |
|---|---|---|
| positive | positive | both negative |
| positive | negative | both positive |
| negative | positive | one positive, one negative (the root with the bigger value, ignoring sign, is negative) |
| negative | negative | one positive, one negative (the root with the bigger value, ignoring sign, is positive) |
Example: x² + 9x + 20 = 0. Here c and b are both positive, so both roots are negative: −4, −5.
How to Compare x and y in Quadratic Equations (Prelims Rules)
- Solve both equations and write all roots of x and y.
- Plot them on a number line, or compare every x with every y.
- Choose the answer using this rule:
| If | Answer |
|---|---|
| every x > every y | x > y |
| every x ≥ every y (with some equal) | x ≥ y |
| every x < every y | x < y |
| every x ≤ every y (with some equal) | x ≤ y |
| the comparison changes between pairs, or all values are equal | x = y or no relation |
Answer options used in the questions below
- (a) x > y
- (b) x ≥ y
- (c) x < y
- (d) x ≤ y
- (e) x = y or no relation can be established
Quadratic Equation Questions with Solutions for SBI PO and IBPS PO (Q1 to Q20)
Section A: Basic factorization (Q1 to Q5)
Directions: Solve both equations and compare x and y.
Q1. I. x² − 7x + 12 = 0
II. y² − 11y + 30 = 0
Solution: I. (x − 3)(x − 4) = 0, so x = 3, 4.
II. (y − 5)(y − 6) = 0, so y = 5, 6. Every x is smaller than every y.
Answer: (c) x < y
Q2. I. x² + 11x + 28 = 0
II. y² + 9y + 20 = 0
Solution: I. (x + 4)(x + 7) = 0, so x = −4, −7.
II. (y + 4)(y + 5) = 0, so y = −4, −5. x = −4 is greater than y = −5, but x = −7 is less than y = −4.
Answer: (e) no relation
Q3. I. x² + 5x + 6 = 0
II. y² − y − 6 = 0
Solution: I. (x + 2)(x + 3) = 0, so x = −2, −3.
II. (y − 3)(y + 2) = 0, so y = 3, −2. x = −2 equals y = −2 and every other pair has x smaller.
Answer: (d) x ≤ y
Q4. I. x² − 15x + 56 = 0
II. y² − 12y + 35 = 0
Solution: I. (x − 7)(x − 8) = 0, so x = 7, 8.
II. (y − 5)(y − 7) = 0, so y = 5, 7. x = 7 equals y = 7 and every other pair has x larger.
Answer: (b) x ≥ y
Q5. I. x² − 5x − 14 = 0
II. y² − 2y − 15 = 0
Solution: I. (x − 7)(x + 2) = 0, so x = 7, −2.
II. (y − 5)(y + 3) = 0, so y = 5, −3. x = 7 is greater than y = 5, but x = −2 is less than y = 5.
Answer: (e) no relation
Section B: When the coefficient of x² is not 1 (Q6 to Q8)
Q6. I. 2x² − 7x + 3 = 0
II. 2y² + 7y + 6 = 0
Solution: I. 2x² − 6x − x + 3 = (2x − 1)(x − 3) = 0, so x = 1/2, 3.
II. 2y² + 4y + 3y + 6 = (2y + 3)(y + 2) = 0, so y = −3/2, −2. Every x is positive and every y is negative.
Answer: (a) x > y
Q7. I. 6x² − 5x − 6 = 0
II. 2y² + 5y + 2 = 0
Solution: I. a × c = −36, and the numbers −9 and +4 give 6x² − 9x + 4x − 6 = (3x + 2)(2x − 3) = 0, so x = −2/3, 3/2.
II. 2y² + 4y + y + 2 = (2y + 1)(y + 2) = 0, so y = −1/2, −2. Now x = −2/3 (about −0.67) is less than y = −1/2 but greater than y = −2.
Answer: (e) no relation
Q8. I. 3x² − 4x − 4 = 0
II. y² − 5y + 6 = 0
Solution: I. 3x² − 6x + 2x − 4 = (3x + 2)(x − 2) = 0, so x = −2/3, 2.
II. (y − 2)(y − 3) = 0, so y = 2, 3. x = 2 equals y = 2 and every other pair has x smaller.
Answer: (d) x ≤ y
Section C: Special cases (Q9 and Q10)
Q9. I. x² − 14x + 49 = 0
II. y² − 15y + 56 = 0
Solution: I. (x − 7)² = 0, so x = 7 (a repeated root).
II. (y − 7)(y − 8) = 0, so y = 7, 8. x = 7 equals y = 7 and is less than y = 8.
Answer: (d) x ≤ y
Q10. I. x² = 9
II. y² + 6y + 9 = 0
Solution: I. x = ±3.
II. (y + 3)² = 0, so y = −3. x = 3 is greater than y = −3 and x = −3 equals y = −3.
Answer: (b) x ≥ y
Section D: Sum and Product of Roots: Practice Questions (Q11 to Q15)
Q11. The roots of x² − 9x + k = 0 differ by 3. Find k.
Solution: Let the roots be α and α + 3. Sum: 2α + 3 = 9, so α = 3. The roots are 3 and 6, and k = product = 18.
Q12. One root of x² − 8x + k = 0 is three times the other. Find k.
Solution: Let the roots be α and 3α. Sum: 4α = 8, so α = 2. The roots are 2 and 6, and k = 12.
Q13. For what value of k does x² + kx + 16 = 0 have equal roots?
Solution: Equal roots need D = 0. k² − 4(1)(16) = 0, so k² = 64 and k = ±8.
Q14. If α and β are the roots of x² − 7x + 10 = 0, find α² + β².
Solution: α + β = 7 and αβ = 10. α² + β² = (α + β)² − 2αβ = 49 − 20 = 29.
Q15. If α and β are the roots of 2x² − 6x + 3 = 0, find α/β + β/α.
Solution: α + β = 6/2 = 3 and αβ = 3/2. α² + β² = 9 − 3 = 6. α/β + β/α = (α² + β²) / αβ = 6 ÷ (3/2) = 4.
Section E: Quadratic Equation Word Problems, Mains Level (Q16 to Q20)
Q16. The product of two consecutive positive integers is 342. Find the integers.
Solution: Let the integers be n and n + 1. n² + n − 342 = 0, so (n − 18)(n + 19) = 0. Take the positive value: n = 18. The integers are 18 and 19.
Q17. The length of a rectangle is 5 m more than its breadth. Its area is 150 m². Find the perimeter.
Solution: Let the breadth be b. b(b + 5) = 150, so b² + 5b − 150 = 0, which gives (b + 15)(b − 10) = 0 and b = 10 m. Length = 15 m. Perimeter = 2(10 + 15) = 50 m.
Q18. The sum of a number and its reciprocal is 10/3. Find the number.
Solution: x + 1/x = 10/3, so 3x² − 10x + 3 = 0. (3x − 1)(x − 3) = 0, so x = 3 or 1/3.
Q19. The product of the ages of two brothers is 238 and the difference between their ages is 3 years. Find their ages.
Solution: Let the ages be a and a + 3. a² + 3a − 238 = 0, so (a + 17)(a − 14) = 0 and a = 14. The ages are 14 and 17 years.
Q20. A boat takes a total of 5 hours to go 24 km upstream and come back 24 km downstream. If the speed of the stream is 2 km/h, find the speed of the boat in still water.
Solution: Let the boat speed be b.
24/(b − 2) + 24/(b + 2) = 5
24 × 2b / (b² − 4) = 5
48b = 5b² − 20
5b² − 48b − 20 = 0
(5b + 2)(b − 10) = 0
Speed cannot be negative, so b = 10 km/h. Check: 24/8 = 3 hours and 24/12 = 2 hours, which total 5 hours.
Common Mistakes in Quadratic Equations
- Sign errors: (x + 3) = 0 gives x = −3, not +3.
- Not bringing everything to one side before factorizing.
- Comparing only one pair of roots. Always compare every x with every y.
- Missing the “equal” case in x ≥ y and x ≤ y questions (see Q3, Q4 and Q10).
- Taking a negative value for age, length or speed in word problems.
Quadratic Equation Tricks to Save Time
- Try factorization for 10 seconds. If nothing clicks, use the formula.
- Use the sign trick to guess the signs of the roots first.
- For sum and product questions, never solve for the roots. Use −b/a and c/a directly.
- For word problems, form the equation carefully. The solving part is easy once the equation is right.
FAQs on Quadratic Equations for Bank Exams
Are quadratic equations asked in SBI PO and IBPS PO?
They have been a regular topic in many bank exams, but the exact pattern changes from year to year. Check the latest exam notification and the previous year’s paper to see how they are being asked now.
What is the best trick to solve quadratic equations quickly?
Splitting the middle term is fastest when the roots are integers, and the sign trick helps you guess the root signs in advance. Use the quadratic formula only when factorization is not visible.
What does “no relation can be established” mean?
It means the comparison between x and y changes depending on which roots you pick, or all values are equal.
Which topics are linked to quadratic equations?
Time and Work, Boats and Streams, Age problems and Area problems often reduce to a quadratic equation.
Download Quadratic Equation Questions PDF
Get the 20 quadratic equation questions with solutions PDF and practice offline with a timer. Aim for under 1 minute per comparison question and 2 minutes per word problem.
👉 [Download Quadratic Equation Questions PDF] (join our Telegram community to get the link)
Revise the squares and cubes table before you start. It makes factorization much faster.
Related posts:
- Squares and Cubes Chart for Bank Exams (Free PDF)
- Approximation and Simplification Tricks for Bank Exams
- Caselet DI: How to Extract Data Fast, with 5 Examples
Next post: Data Sufficiency for IBPS PO Mains: 10 Questions and the Elimination Method
