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Algebra

Mathematical Abilities — Paper-I · Official 2026 Coverage

Useful for: SSC CGL

Overview

Tier-II algebra uses identities, quadratic equations and symmetric expressions in two variables.

Key points

a³ + b³ = (a + b)³ − 3ab(a + b).
x² + y² = (x − y)² + 2xy = (x + y)² − 2xy.
For ax² + bx + c = 0: sum of roots = −b/a, product = c/a.
Discriminant b² − 4ac: positive → two real roots; zero → equal roots; negative → no real roots.
If a + b + c = 0, then a³ + b³ + c³ = 3abc.

Explanation

Form a quadratic from roots α and β: x² − (α + β)x + αβ = 0.
Use substitution of simple values to test identity-based options.
For x + 1/x type expressions, square and cube step by step.
Linear equations in two variables: check consistency using coefficient ratios.

Step by step

Identify the concept and the formula that links the given values.
Solve step by step, keeping units consistent.
Check the answer against the options and by a quick estimate.

Examples

a + b = 5, ab = 6 → a³ + b³ = 125 − 3 × 6 × 5 = 35.
x − y = 3, xy = 10 → x² + y² = 9 + 20 = 29.
x² − 5x + 6 = 0 → roots 2 and 3; sum 5, product 6.

Traps and advanced points

Trap: Sign errors in −b/a.
Trap: Using (a + b)² in place of (a + b)³ identities.
Trap: Forgetting the 2xy term.

Quick revision

Express in a + b and ab.
Sum −b/a, product c/a.
Discriminant decides roots.

Practice MCQs

If 4x + 12 = 44, find x.
1. 10
2. 8
3. 7
4. 9
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Answer: (b) 8. 4x=32; x=8.
Solve: 4x + 15 = 63.
1. 14
2. 10
3. 16
4. 12
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Answer: (d) 12. 4x=48.
Solve: 4x + 7 = 55.
1. 14
2. 16
3. 12
4. 10
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Answer: (c) 12. 4x=48.
For x² − 25x + 156 = 0, what is the sum of its roots?
1. 23
2. 27
3. 29
4. 25
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Answer: (d) 25. For ax²+bx+c=0, sum of roots = −b/a.
Solve: 5x + 13 = 78.
1. 17
2. 11
3. 15
4. 13
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Answer: (d) 13. 5x=65.
Solve: 3x + 5 = 47.
1. 18
2. 12
3. 14
4. 16
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Answer: (c) 14. Correct answer: 14. Apply the core rule for Use identities such as (a+b)^2, (a−b)^2 and a^2−b^2.
Solve: 4x + 13 = 49.
1. 11
2. 7
3. 13
4. 9
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Answer: (d) 9. Correct answer: 9. Apply the core rule for Use identities such as (a+b)^2, (a−b)^2 and a^2−b^2.
For x² − 9x + 20 = 0, what is the sum of its roots?
1. 13
2. 7
3. 9
4. 11
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Answer: (c) 9. For ax²+bx+c=0, sum of roots = −b/a.
Solve: 7x + 8 = 148.
1. 20
2. 22
3. 24
4. 18
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Answer: (a) 20. Correct answer: 20. Apply the core rule for Use identities such as (a+b)^2, (a−b)^2 and a^2−b^2.
For x² − 11x + 30 = 0, what is the sum of its roots?
1. 11
2. 13
3. 9
4. 15
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Answer: (a) 11. For ax²+bx+c=0, sum of roots = −b/a.
Solve: 4x + 11 = 47.
1. 7
2. 13
3. 9
4. 11
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Answer: (c) 9. Correct answer: 9. Apply the core rule for Use identities such as (a+b)^2, (a−b)^2 and a^2−b^2.
For x² − 13x + 42 = 0, what is the sum of its roots?
1. 13
2. 15
3. 11
4. 17
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Answer: (a) 13. For ax²+bx+c=0, sum of roots = −b/a.
Solve: 5x + 4 = 29.
1. 3
2. 9
3. 5
4. 7
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Answer: (c) 5. Correct answer: 5. Apply the core rule for Use identities such as (a+b)^2, (a−b)^2 and a^2−b^2.
For x² − 15x + 56 = 0, what is the sum of its roots?
1. 19
2. 13
3. 15
4. 17
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Answer: (c) 15. For ax²+bx+c=0, sum of roots = −b/a.
Solve: 5x + 13 = 33.
1. 8
2. 4
3. 2
4. 6
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Answer: (b) 4. Correct answer: 4. Apply the core rule for Use identities such as (a+b)^2, (a−b)^2 and a^2−b^2.