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Quadratic Equations

Quantitative Aptitude / DI · Speed Calculation

Useful for: IBPS Banking

Overview

IBPS quadratic questions ask you to solve two equations and state the relation between x and y.

Key points

Factorise x² + bx + c by finding two numbers with product c and sum b.
Signs: if b and c are positive, both roots are negative.
If c is negative, the roots have opposite signs.
Compare all pairs: x > y only if every x is greater than every y.
If one equation is x² = k, roots are ±√k.

Explanation

Equal roots in both sets give ≥ or ≤.
If pairs give mixed results, the relation cannot be determined.
Use the sign rule to save time before factorising.
For ax² + bx + c with a ≠ 1, split the middle term using a × c.

Step by step

Translate the data or statement into numbers.
Compute step by step with the right formula.
Compare with options; check reasonableness.

Examples

x² − 5x + 6 = 0 → x = 2, 3; y² − 7y + 12 = 0 → y = 3, 4 → x ≤ y.
x² = 16 → x = ±4; y = 4 → x ≤ y.
x² + 7x + 12 = 0 → x = −3, −4; y² + 5y + 6 = 0 → y = −2, −3 → x ≤ y.

Traps and advanced points

Trap: Comparing only one pair of roots.
Trap: Sign errors in roots.
Trap: Forgetting the negative root of x² = k.

Quick revision

Product c, sum b.
Compare all pairs.
Mixed → cannot be determined.

Practice MCQs

Roots of x² − 5x + 6 = 0 are —
1. 1 and 6
2. −1 and 6
3. 2 and 3
4. −2 and −3
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Answer: (c) 2 and 3. (x − 2)(x − 3) = 0.
Roots of x² + 7x + 12 = 0 are —
1. 3 and 4
2. −3 and −4
3. −3 and 4
4. 3 and −4
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Answer: (b) −3 and −4. (x + 3)(x + 4) = 0.
If b and c are both positive in x² + bx + c = 0, the roots are —
1. both positive
2. both negative
3. of opposite signs
4. zero
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Answer: (b) both negative. Sum −b, product c.
If c is negative in x² + bx + c = 0, the roots are —
1. both negative
2. of opposite signs
3. equal
4. both positive
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Answer: (b) of opposite signs. Product negative.
Roots of x² = 16 are —
1. 4 only
2. ±4
3. 16
4. ±8
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Answer: (b) ±4. ±√16.
I: x² − 7x + 12 = 0, II: y² − 3y + 2 = 0. Relation —
1. cannot be determined
2. x > y
3. x < y
4. x ≥ y
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Answer: (b) x > y. x ∈ {3, 4}, y ∈ {1, 2}.
I: x² = 9, II: y = 3. Relation —
1. x = y
2. x > y
3. x < y
4. x ≤ y
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Answer: (d) x ≤ y. x ∈ {−3, 3}; both ≤ 3.
Sum of roots of x² − 5x + 6 = 0 is —
1. 6
2. −6
3. −5
4. 5
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Answer: (d) 5. −b/a = 5.
Roots of 2x² − 5x + 2 = 0 are —
1. 2 and 1/2
2. −2 and −1/2
3. 1 and 2
4. 5 and 2
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Answer: (a) 2 and 1/2. (2x − 1)(x − 2) = 0.
I: x² − 1 = 0, II: y² + 2y + 1 = 0. Relation —
1. x ≥ y
2. x < y
3. x = y
4. x > y
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Answer: (a) x ≥ y. x ∈ {−1, 1}, y = −1.