Quadratic Equations
Overview
Quadratic equations in IBPS Clerk Prelims follow a fixed, predictable pattern: you receive two equations (one in x, one in y), solve both, then compare the roots to determine the relationship between x and y.
Unlike school-level quadratic problems, the exam doesn't ask you to find roots for their own sake. Your job is to compare values: Is x greater than y? Are they equal? Is the relationship indeterminate? The equations are designed to have integer roots (usually between -10 and +10), so factorization works smoothly without needing the quadratic formula.
Students who drill this topic properly can finish all 5 questions in under 3 minutes—a significant advantage when time pressure is intense in the 60-question, 60-minute Prelims paper.
Key Concepts
- Standard form: A quadratic equation is written as ax² + bx + c = 0, where a, b, c are constants and a ≠ 0.
- Roots: The values of x (or y) that satisfy the equation. Each quadratic has exactly two roots (which may be equal or distinct).
- Sum of roots = -b/a; Product of roots = c/a. Use these to verify your factorization quickly.
- Factorization method: Split the middle term into two parts whose product equals a × c and whose sum equals b.
- Sign convention for comparison: After finding both roots of each equation, compare all possible pairs and decide the universal relationship.
- Answer options in IBPS exams typically are:
- x > y (all values of x are greater than all values of y)
- x < y
- x ≥ y (x is greater than or equal to y in all cases)
- x ≤ y
- x = y or relationship cannot be established
- "Cannot be established": If some pairs give x > y and others give x < y, the relationship is indeterminate.
Formulas / Key Facts
| Formula / Fact | Context |
|---|---|
| ax² + bx + c = 0 | Standard quadratic form |
| Sum of roots = -b/a | Quick verification after factorization |
| Product of roots = c/a | Helps confirm root signs (+/−) |
| Discriminant = b² - 4ac | If negative, no real roots (rare in Clerk exams) |
| (x - p)(x - q) = 0 means roots are p and q | Factorization outcome |
| For comparison: arrange roots on a number line | Visual clarity for relationship |
IBPS-specific fact: Equations are crafted so roots are small integers. If you're getting messy decimals, recheck your factorization.
Worked Examples
Example 1
Equation I: x² - 7x + 12 = 0 Equation II: y² - 9y + 20 = 0
Step 1: Solve for x Find two numbers that multiply to 12 and add to 7: 3 and 4 x² - 3x - 4x + 12 = 0 x(x - 3) - 4(x - 3) = 0 (x - 3)(x - 4) = 0 x = 3 or x = 4
Step 2: Solve for y Find two numbers that multiply to 20 and add to 9: 4 and 5 y² - 4y - 5y + 20 = 0 y(y - 4) - 5(y - 4) = 0 (y - 4)(y - 5) = 0 y = 4 or y = 5
Step 3: Compare x values: 3, 4 y values: 4, 5
| Comparison | Result |
|---|---|
| 3 vs 4 | x < y |
| 3 vs 5 | x < y |
| 4 vs 4 | x = y |
| 4 vs 5 | x < y |
Answer: x ≤ y
Example 2
Equation I: x² + 5x + 6 = 0 Equation II: y² + 7y + 10 = 0
Step 1: Solve for x Factors of 6 that add to 5: 2 and 3 (x + 2)(x + 3) = 0 x = -2 or x = -3
Step 2: Solve for y Factors of 10 that add to 7: 2 and 5 (y + 2)(y + 5) = 0 y = -2 or y = -5
Step 3: Compare x values: -2, -3 y values: -2, -5
| Comparison | Result |
|---|---|
| -2 vs -2 | x = y |
| -2 vs -5 | x > y (since -2 > -5) |
| -3 vs -2 | x < y |
| -3 vs -5 | x > y |
Some pairs give x > y, others give x < y. Answer: Relationship cannot be established
Example 3
Equation I: 2x² - 7x + 3 = 0 Equation II: 2y² - 7y - 4 = 0
Step 1: Solve for x a × c = 2 × 3 = 6 Find two numbers that multiply to 6 and add to -7: -6 and -1 2x² - 6x - x + 3 = 0 2x(x - 3) - 1(x - 3) = 0 (2x - 1)(x - 3) = 0 x = 0.5 or x = 3
Step 2: Solve for y a × c = 2 × (-4) = -8 Find two numbers that multiply to -8 and add to -7: -8 and +1 2y² - 8y + y - 4 = 0 2y(y - 4) + 1(y - 4) = 0 (2y + 1)(y - 4) = 0 y = -0.5 or y = 4
Step 3: Compare x values: 0.5, 3 y values: -0.5, 4
| Comparison | Result |
|---|---|
| 0.5 vs -0.5 | x > y |
| 0.5 vs 4 | x < y |
| 3 vs -0.5 | x > y |
| 3 vs 4 | x < y |
Mixed results. Answer: Relationship cannot be established
Common Mistakes
- Ignoring negative roots in comparison: Students forget that -3 > -5. Fix: Draw a number line mentally; numbers increase as you move right.
- Assuming single overlap means x = y: If x = 4 and y = 4 appear, but other values differ, the answer is ≥ or ≤, not strictly equal. Fix: Compare ALL root pairs.
- Sign errors during factorization: When c is positive and b is negative, both factors are negative. When c is negative, factors have opposite signs. Fix: Check product and sum before finalizing.
- Rushing to "cannot be established": Some students pick this option too quickly. Fix: Systematically compare all four pairs (2 x-values × 2 y-values).
- Forgetting to divide when a ≠ 1: For 2x² equations, after factorizing to (2x - 1) = 0, the root is x = 1/2, not x = 1. Fix: Always solve the bracket completely.
Quick Reference
- Factorize both equations → Get two roots each → Compare all four pairs
- Sum of roots = -b/a, Product of roots = c/a (use to verify)
- All x > all y → Answer: x > y
- All x < all y → Answer: x < y
- Some equal, rest one-sided → Answer: x ≥ y or x ≤ y
- Mixed results → Answer: Cannot be established
- Typical root range in IBPS Clerk: integers from -10 to +10