Questions
3 questions per paper
Difficulty
Easy
Importance
High yield for banking and SSC prelims
Overview
Quadratic comparison is a high-frequency scoring area in competitive exams involving the relational analysis of roots between two equations. Aspirants must master quick sign-based elimination tactics to identify the relationship between variables X and Y without fully solving the equations.
Standard Quadratic Form and Roots
Every quadratic equation follows the standard form ax^2 + bx + c = 0. Mastery involves finding roots x1 and x2 using the quadratic formula or factorization, which is the foundational step for all comparison-based problems.
- Standard form: ax^2 + bx + c = 0
- Quadratic formula: x = [-b ± sqrt(b^2 - 4ac)] / 2a
- Sum of roots: -b/a
- Product of roots: c/a
- Nature of roots determined by Discriminant (D) = b^2 - 4ac
Speed-Solving via Sign Rules
In many competitive exam questions, you can determine the relationship between roots just by looking at the signs of coefficients. This technique eliminates the need for full calculation and saves critical seconds.
- If c is negative, the roots always have opposite signs
- If c is positive and b is negative, both roots are positive
- If c is positive and b is positive, both roots are negative
- When c < 0 in both equations, the relationship is often 'cannot be determined'
- Always compare each root of X with each root of Y
Quantity Comparison Strategy
Quantity I vs. Quantity II problems require evaluating two distinct mathematical expressions and determining which is larger. Treat each quantity as a separate calculation task before applying comparative operators.
- Solve Quantity I independently
- Solve Quantity II independently
- Assign relational operators: >, <, ≥, ≤, =
- Use 'No Relation' (CND) if the root intervals overlap
- Check for common roots to simplify expressions
Formula Sheet
ax^2 + bx + c = 0
x = [-b ± sqrt(b^2 - 4ac)] / 2a
Sum of roots (α + β) = -b/a
Product of roots (αβ) = c/a
D = b^2 - 4ac
Exam Tip
If both equations have a negative constant term (c < 0), the answer is almost always 'Relationship cannot be established'—mark and move on immediately.
Common Mistakes
- Ignoring the sign of the coefficient 'a' when 'a' is not equal to 1, leading to incorrect root signs.
- Forgetting that 'Cannot be determined' (CND) is a valid and frequent answer when root ranges overlap.
- Spending too much time calculating exact values when sign analysis is sufficient to eliminate options.
More Revision Notes
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