Quadratic Equations FOR Placement

Meta Description: Master quadratic equations for placement exams 2026. Learn formulas, solving tricks, nature of roots, and practice 23+ solved problems for banking & IT tests.

Introduction

Quadratic Equations form the backbone of the Quantitative Aptitude section in almost every major placement exam in 2026. In banking exams like IBPS PO, SBI Clerk, and RBI Grade B, candidates typically encounter 4–5 direct questions in prelims, often focusing on comparing roots of two equations. In IT corporate placements (TCS NQT, Infosys, Wipro, Accenture), 1–2 conceptual questions test your speed and accuracy with discriminants or Vieta’s relationships. Mastering this topic can significantly boost your sectional score, as questions are highly predictable and formula-driven. This guide covers every essential concept, shortcut, and exam-level problem to ensure you solve quadratic equations confidently and quickly.

Key Formulas & Concepts

Standard Form:
ax² + bx + c = 0 where a ≠ 0

1. Quadratic Formula:
x = [-b ± √(b² - 4ac)] / 2a
Gives exact roots for any quadratic equation.

2. Discriminant (D):
D = b² - 4ac

• D > 0 → Two distinct real roots
• D = 0 → Two equal real roots
• D < 0 → Complex/imaginary roots (no real solution)

3. Vieta’s Formulas (Root Relationships):
If α and β are roots:

• Sum: α + β = -b/a
• Product: αβ = c/a

4. Factorization Method:
Split the middle term bx into two terms whose product equals a × c and sum equals b.

5. Root Comparison Rule:
Solve both equations, list roots in ascending order, then compare every possible pair. If all combinations yield the same inequality, establish a relation. Otherwise, relation cannot be established.

Solved Examples (Basic Level)

1. Solve: 2x² - 7x + 3 = 0
Product = a×c = 6, Sum = -7. Split: -6x - x
2x² - 6x - x + 3 = 0 → 2x(x - 3) - 1(x - 3) = 0
(2x - 1)(x - 3) = 0 → x = 1/2, 3

2. Solve: x² - 5x + 6 = 0
Split: -3x - 2x → x(x - 3) - 2(x - 3) = 0
(x - 3)(x - 2) = 0 → x = 3, 2

3. Use quadratic formula: 3x² + 2x - 1 = 0
a=3, b=2, c=-1, D = 4 + 12 = 16
x = [-2 ± √16]/6 → x = (-2+4)/6 = 1/3, x = (-2-4)/6 = -1

4. Find nature of roots: 4x² + 4x + 1 = 0
D = 16 - 16 = 0 → Roots are real and equal.

5. Find equation if sum of roots = -3, product = -10
Standard form: x² - (sum)x + product = 0
x² + 3x - 10 = 0

Practice Questions (Medium Level)

6. Compare: I. 2x² - 9x + 10 = 0 | II. 4y² - 13y + 10 = 0
I: (2x - 5)(x - 2) = 0 → x = 5/2, 2
II: (4y - 5)(y - 2) = 0 → y = 5/4, 2
Comparing: 2.5 ≥ 1.25, 2.5 ≥ 2, 2 ≥ 1.25, 2 = 2 → x ≥ y

7. If α, β are roots of x² - 8x + 15 = 0, find 1/α + 1/β
α + β = 8, αβ = 15
1/α + 1/β = (α + β)/αβ = 8/15

8. Solve: √3x² + 10x + 7√3 = 0
a=√3, b=10, c=7√3, D = 100 - 84 = 16
x = [-10 ± 4]/(2√3) → x = -6/(2√3) = -√3, x = -14/(2√3) = -7/√3

9. Find k if roots of 2x² + kx + 18 = 0 are equal
Equal roots → D = 0 → k² - 4(2)(18) = 0 → k² = 144 → k = ±12

10. Compare: I. 6x² + 7x + 2 = 0 | II. 15y² - 38y - 40 = 0
I: (3x + 2)(2x + 1) = 0 → x = -2/3, -1/2
II: (5y - 20)(3y + 2) → (5y + 4)(3y - 10) = 0 → y = -4/5, 10/3
-0.66 > -0.8, -0.5 > -0.8, but -0.66 < 3.33 → Mixed → Relation cannot be established

11. One root of 3x² - 5x + p = 0 is 2. Find p & other root.
Substitute x=2: 12 - 10 + p = 0 → p = -2
Equation: 3x² - 5x - 2 = 0 → (3x + 1)(x - 2) = 0 → Other root = -1/3

12. Form quadratic with roots (2 + √5) and (2 - √5)
Sum = 4, Product = 4 - 5 = -1 → x² - 4x - 1 = 0

13. Solve: x² - (√2 + 1)x + √2 = 0
Split: -√2x - x → x(x - √2) - 1(x - √2) = 0 → x = 1, √2

Advanced Questions

14. If α, β are roots of x² - 2x + 3 = 0, find α² + β²
α + β = 2, αβ = 3
α² + β² = (α + β)² - 2αβ = 4 - 6 = -2

15. Compare: I. x² - 7x + 12 = 0 | II. y² - 5y + 6 = 0
I: x = 3, 4 | II: y = 2, 3
3 ≥ 2, 4 > 2, 3 = 3, 4 > 3 → All satisfy x ≥ y → x ≥ y

16. Find range of m for which x² + (m+1)x + 1 = 0 has real roots
Real roots → D ≥ 0 → (m+1)² - 4 ≥ 0 → (m+1)² ≥ 4
m+1 ≥ 2 or m+1 ≤ -2 → m ≥ 1 or m ≤ -3

17. If roots are in ratio 2:3, prove 6b² = 25ac
Let roots be 2k, 3k. Sum = 5k = -b/a → k = -b/5a
Product = 6k² = c/a → Substitute k: 6(b²/25a²) = c/a
Multiply by 25a²: 6b² = 25ac → Proved

18. Solve: (x + 2)/(x - 2) + (x - 2)/(x + 2) = 10/3
Let u = (x+2)/(x-2). Then u + 1/u = 10/3
3u² - 10u + 3 = 0 → (3u - 1)(u - 3) = 0 → u = 1/3, 3
Case 1: (x+2)/(x-2) = 3 → x = 4
Case 2: (x+2)/(x-2) = 1/3 → x = -4 → x = ±4

Common Mistakes to Avoid

• Ignoring the negative sign in sum formula: α + β = -b/a (not +b/a)
• Declaring D < 0 as "no solution" instead of "no real roots/imaginary roots"
• Forgetting to check both + and - when using the quadratic formula
• Incorrectly establishing x > y when roots overlap (e.g., x = {3,5}, y = {2,4} → relation undefined)
• Assuming a = 1 always, leading to wrong factorization splits
• Missing the condition a ≠ 0 in parameter-based questions

Shortcut Tricks

1. Sum-Product Quick Factorization:
Instead of trial & error, find two numbers whose product is a×c and sum is b. Divide by a if needed. Example: 6x² + 7x + 2 = 0 → 12 & 7 → split directly.

2. Discriminant Sign Check (Mental):
For ax² + bx + c, if a & c have same sign, b² must exceed 4ac for real roots. If opposite signs, D is always positive → always real roots.

3. Reciprocal Roots Shortcut:
If roots are reciprocals, c/a = 1 → c = a. Saves time in MCQs.

4. Comparison Table Method:
Arrange roots on a number line: y1 y2 x1 x2 → instantly see overlaps. If entire set of x is right of y, x > y.

5. Equal Roots Parameter Trick:
When asked "roots are equal", directly apply b² = 4ac. Skip full equation solving.

Previous Year Questions

19. (IBPS PO 2023) Compare: I. 3x² - 13x + 12 = 0 | II. 2y² - 15y + 25 = 0
I: (3x - 4)(x - 3) = 0 → x = 4/3, 3
II: (2y - 5)(y - 5) = 0 → y = 2.5, 5
1.33 < 2.5, 3 < 5, but 3 > 2.5 → Mixed → Relation cannot be established

20. (SBI Clerk 2022) If α + β = 10, αβ = 21, find α³ + β³
Formula: α³ + β³ = (α + β)³ - 3αβ(α + β)
= 1000 - 3(21)(10) = 1000 - 630 = 370

21. (TCS NQT 2023) Find roots of 2x² - √5x - 3 = 0
D = 5 + 24 = 29
x = [√5 ± √29]/4 → x = (√5 + √29)/4, (√5 - √29)/4

22. (Infosys 2022) For what k are roots of x² - 4x + k = 0 real and distinct?
Distinct real → D > 0 → 16 - 4k > 0 → 4k < 16 → k < 4

23. (RBI Grade B 2021) Compare: I. x² + 8x + 15 = 0 | II. 2y² + 11y + 14 = 0
I: x = -3, -5
II: y = -2, -3.5
-3 < -2, -3 > -3.5, -5 < -3.5, -5 < -2 → Overlap at different positions → Relation cannot be established

Quick Revision

• Standard form: ax² + bx + c = 0, a ≠ 0
• Roots: x = [-b ± √(b²-4ac)]/2a
• Discriminant D = b² - 4ac dictates nature (>, =, < 0)
• Sum of roots = -b/a, Product = c/a
• For comparison: solve both, list roots, test all 4 pairs
• Equal roots → D = 0; Real roots → D ≥ 0
• Always verify sign of b in -b/a
• Banking exams prefer comparison questions; IT tests focus on discriminant & parameters
• Practice 10–15 comparison sets daily for speed under 90 seconds per set

Frequently Asked Questions

What salary range can candidates expect after clearing placement exams that include quadratic equations?

Salary outcomes depend more on the overall placement score and the final company/role than on a single topic like quadratic equations. However, candidates who consistently perform well in Quantitative Aptitude (including quadratic equations) typically improve their chances of clearing sectional cutoffs, which can lead to better offers in banking and IT roles.

What is the eligibility required to attempt placement exams where quadratic equations are asked?

Most placement and banking exams require a minimum educational qualification (commonly graduation) and basic eligibility criteria like age limits and nationality, which vary by exam. For the Quant section, there’s usually no special eligibility beyond having standard school-level algebra knowledge (polynomials, factoring, and basic algebraic manipulation).

How difficult are quadratic equations questions in placement exams 2026?

Quadratic equations questions are generally moderate in difficulty, but they can become tricky due to word problems, parameter-based questions, and questions on the nature of roots. The difficulty usually comes from choosing the fastest method, factorization, completing the square, or using the discriminant (D = b² − 4ac) - under time pressure.

What are the best preparation tips to score high in quadratic equations for placement exams?

Start by mastering the core formulas (roots, discriminant, and relationships between roots) and then practice a variety of question types like finding roots, nature of roots, and parameter constraints. Use time-saving tricks such as checking discriminant sign quickly and using Vieta’s formulas (sum and product of roots) to avoid lengthy calculations.

How do quadratic equations typically appear across interview rounds or selection stages?

In most placement processes, quadratic equations are part of the written/online aptitude test rather than the final HR interview. Some companies may include quant reasoning questions in the technical round, but the majority of quadratic-equation practice is aimed at clearing the written test and sectional cutoffs.

Which common topics from quadratic equations are frequently asked in banking & IT placement exams?

Common topics include solving quadratic equations by factorization or formula method, determining the nature of roots using the discriminant, and solving equations involving parameters. You’ll also see questions on forming equations from conditions, finding values of parameters for real roots, and applying root relationships (sum/product) to simplify problems.

How can I apply or enroll to practice quadratic equations for placement exams 2026?

You can typically apply by registering on the placement preparation platform that provides the quadratic-equations practice set, formulas, and solved problems. Look for sections like “Formulas & Tricks” and “Practice Problems (23+ solved)” and follow a structured plan: learn → practice solved examples → attempt timed mixed sets.

What is the expected selection rate for candidates who prepare quadratic equations well?

There isn’t a single universal selection rate because it varies by exam, competition level, and overall performance across all sections. However, strong preparation in quadratic equations improves accuracy in Quantitative Aptitude, which can significantly raise your probability of clearing sectional cutoffs and reaching later stages.