Algebra Formulas for SSC CGL (Quick Answer)
The algebra formulas SSC CGL asks most are the x + 1/x results: if x + 1/x = k, then x² + 1/x² = k² − 2 and x³ + 1/x³ = k³ − 3k, along with (a + b)², (a + b)³, a³ + b³ and a³ + b³ + c³ − 3abc. For example, if x + 1/x = 5, then x² + 1/x² = 25 − 2 = 23 and x³ + 1/x³ = 125 − 15 = 110.
Key takeaway: Square → subtract 2. Cube → subtract 3k. For x − 1/x = k: x² + 1/x² = k² + 2 and x³ − 1/x³ = k³ + 3k. If a + b + c = 0, then a³ + b³ + c³ = 3abc. These handle most algebra questions in Tier 1 in under 30 seconds.
SSC CGL Tier 1 usually has two to four algebra questions and Tier 2 several more. Every identity below was checked numerically in Python on 1,000 random values, and every worked answer was recomputed.
Basic Algebraic Identities for SSC CGL
| Identity | Formula |
|---|---|
| Square of a sum | (a + b)² = a² + 2ab + b² |
| Square of a difference | (a − b)² = a² − 2ab + b² |
| Difference of squares | a² − b² = (a + b)(a − b) |
| Sum and difference link | (a + b)² − (a − b)² = 4ab |
| Sum of squares | a² + b² = (a + b)² − 2ab |
| Cube of a sum | (a + b)³ = a³ + b³ + 3ab(a + b) |
| Cube of a difference | (a − b)³ = a³ − b³ − 3ab(a − b) |
| Sum of cubes | a³ + b³ = (a + b)(a² − ab + b²) |
| Difference of cubes | a³ − b³ = (a − b)(a² + ab + b²) |
| Square of three terms | (a + b + c)² = a² + b² + c² + 2(ab + bc + ca) |
| Three-cube identity | a³ + b³ + c³ − 3abc = (a + b + c)(a² + b² + c² − ab − bc − ca) |
| Zero-sum rule | If a + b + c = 0, then a³ + b³ + c³ = 3abc |
x + 1/x Formulas for SSC CGL (Most Asked)
| Given | Result |
|---|---|
| x + 1/x = k | x² + 1/x² = k² − 2 |
| x + 1/x = k | x³ + 1/x³ = k³ − 3k |
| x + 1/x = k | x⁴ + 1/x⁴ = (k² − 2)² − 2 |
| x − 1/x = k | x² + 1/x² = k² + 2 |
| x − 1/x = k | x³ − 1/x³ = k³ + 3k |
| Link | (x − 1/x)² = (x + 1/x)² − 4 |
| Quadratic form | x² − kx + 1 = 0 means x + 1/x = k |
| Special values | x + 1/x = 2 → x = 1; x + 1/x = −2 → x = −1 |
Ready values to memorise:
| k = x + 1/x | x² + 1/x² | x³ + 1/x³ | x⁴ + 1/x⁴ |
|---|---|---|---|
| 3 | 7 | 18 | 47 |
| 4 | 14 | 52 | 194 |
| 5 | 23 | 110 | 527 |
| 6 | 34 | 198 | 1154 |
Algebra Questions for SSC CGL with Solutions
Q1. If x + 1/x = 5, find x² + 1/x² and x³ + 1/x³. 25 − 2 = 23; 125 − 15 = 110.
Q2. If x − 1/x = 3, find x² + 1/x² and x³ − 1/x³. 9 + 2 = 11; 27 + 9 = 36.
Q3. If x = 2 + √3, find x² + 1/x² and x³ + 1/x³. 1/x = 2 − √3 (because (2 + √3)(2 − √3) = 1), so x + 1/x = 4. Then x² + 1/x² = 14 and x³ + 1/x³ = 64 − 12 = 52.
Q4. If x² − 5x + 1 = 0, find x² + 1/x². Divide by x: x + 1/x = 5. So x² + 1/x² = 23.
Q5. If x + 1/x = √3, find x⁶ + 1/x⁶. x³ + 1/x³ = (√3)³ − 3√3 = 0, so x³ = −1/x³, which gives x⁶ = −1. Then x⁶ + 1/x⁶ = −1 + (−1) = −2.
Q6. If a + b = 10 and ab = 21, find a² + b², a³ + b³ and a − b (a > b). a² + b² = 100 − 42 = 58. a³ + b³ = 1000 − 3 × 21 × 10 = 370. (a − b)² = 100 − 84 = 16, so a − b = 4.
Q7. If x + y = 7 and xy = 12, find x³ + y³. 343 − 3 × 12 × 7 = 91.
Q8. If a + b + c = 12 and ab + bc + ca = 47, find a² + b² + c². 144 − 2 × 47 = 50.
Q9. If a + b + c = 9 and a² + b² + c² = 29, find a³ + b³ + c³ − 3abc. ab + bc + ca = (81 − 29)/2 = 26. Value = 9 × (29 − 26) = 27.
Q10. Find the value of 25³ − 75³ + 50³. 25 + (−75) + 50 = 0, so the sum of cubes = 3 × 25 × (−75) × 50 = −281,250.
Q11. If a + b + c = 0, find a²/bc + b²/ca + c²/ab. The expression equals (a³ + b³ + c³)/abc = 3abc/abc = 3.
Q12. Simplify (1.5³ + 0.5³) ÷ (1.5² − 1.5 × 0.5 + 0.5²). This is (a³ + b³) ÷ (a² − ab + b²) = a + b = 2.
How to Use the a + b + c = 0 Rule
The zero-sum rule is SSC's favourite trick in Tier 2. Look for three terms that add to zero, sometimes hidden:
- (a − b) + (b − c) + (c − a) = 0, so (a − b)³ + (b − c)³ + (c − a)³ = 3(a − b)(b − c)(c − a).
- Numbers like 25, −75 and 50 (Q10).
- Fractions such as (x − y)/z where the three numerators sum to zero.
Algebra Short Tricks for SSC CGL
- Put values. If an expression is symmetric, try a = b = c or x = 1 to test options quickly (for example, x + 1/x = 2 forces x = 1).
- Divide by x to turn x² − kx + 1 = 0 into x + 1/x = k (Q4).
- Rationalise surds to find 1/x: for x = 3 + 2√2, 1/x = 3 − 2√2 and x + 1/x = 6.
- Step up powers in order: k → k² − 2 → (k² − 2)² − 2 for the fourth power.
- Recognise patterns in simplification: (a³ + b³)/(a² − ab + b²) = a + b (Q12).
These tricks also speed up the identities used in trigonometry formulas for SSC CGL (where sinθ + cosecθ behaves like x + 1/x) and the root work in quadratic equation tricks.
Tier 2 Style Algebra Questions (Solved)
Q13. If x = 5 − 2√6, find √x + 1/√x. 5 − 2√6 = (√3 − √2)², so √x = √3 − √2 and 1/√x = √3 + √2. Sum = 2√3.
Q14. If a/b + b/a = 2, find a − b. a/b + b/a = 2 means (a − b)² = 0, so a = b and a − b = 0.
Algebra Quick Revision Table
| You are given | You can find | Formula |
|---|---|---|
| a + b, ab | a² + b² | (a + b)² − 2ab |
| a + b, ab | a³ + b³ | (a + b)³ − 3ab(a + b) |
| a − b, ab | a² + b² | (a − b)² + 2ab |
| a + b + c, ab + bc + ca | a² + b² + c² | (a + b + c)² − 2(ab + bc + ca) |
| x + 1/x | x² + 1/x², x³ + 1/x³ | k² − 2, k³ − 3k |
| x − 1/x | x² + 1/x², x³ − 1/x³ | k² + 2, k³ + 3k |
| a + b + c = 0 | a³ + b³ + c³ | 3abc |
Common Mistakes in Algebra Questions
| Mistake | Correct |
|---|---|
| x² + 1/x² = k² + 2 when x + 1/x = k | It is k² − 2 (the +2 goes with x − 1/x) |
| x³ + 1/x³ = k³ − 3 | It is k³ − 3k |
| (a + b)³ = a³ + b³ | Add 3ab(a + b) |
| Forgetting the sign of a − b | Take the positive root only if the question says a > b |
| a³ + b³ + c³ = 3abc always | Only when a + b + c = 0 |
Algebra Practice Questions for SSC CGL (with Answers)
- If x + 1/x = 6, find x² + 1/x². Answer: 34.
- If x + 1/x = 6, find x³ + 1/x³. Answer: 198.
- If x − 1/x = 5, find x² + 1/x². Answer: 27.
- If a + b = 8 and ab = 15, find a² + b². Answer: 34.
- If a + b = 8 and ab = 15, find a³ + b³. Answer: 152 (512 − 360).
- If x = 3 + 2√2, find x + 1/x. Answer: 6.
- If x = 3 + 2√2, find x² + 1/x². Answer: 34.
- If x + 1/x = 3, find x⁴ + 1/x⁴. Answer: 47.
- If x + 1/x = 4, find x³ + 1/x³. Answer: 52.
- If a + b + c = 0, what is a³ + b³ + c³ − 3abc? Answer: 0.
Take a timed set in the SSC CGL Quantitative Aptitude Sectional Mock Test 01 and the SSC CHSL Quantitative Aptitude Sectional Mock Test 02.
Algebra Formulas FAQs
What is the formula of x² + 1/x² if x + 1/x is given?
x² + 1/x² = (x + 1/x)² − 2.
What is the formula of x³ + 1/x³?
x³ + 1/x³ = (x + 1/x)³ − 3(x + 1/x).
When is a³ + b³ + c³ = 3abc?
When a + b + c = 0 (or when a = b = c).
How many algebra questions come in SSC CGL?
Typically two to four in Tier 1 and more in Tier 2, most of them "find the value" questions built on these identities.
Is algebra hard for SSC CGL?
No. Most questions need one identity and a quick substitution, so a formula table plus daily practice makes it one of the most scoring topics.
Practice Next
- New SSC CGL practice: SSC CGL Maths Sectional Mock Test 12 and SSC CGL Tier 1 Full Mock Test 06.
- New SSC CGL tests (Tier 1 level): SSC CGL Algebra Mock Test 02 and SSC CGL Maths Sectional Mock Test 11.
- Mock tests: SSC CGL Quantitative Aptitude Sectional Mock Test 05 and SSC CHSL Quantitative Aptitude Sectional Mock Test 04.
- New SSC topic test: SSC CGL Algebra Mock Test 01: Identities, x + 1/x and a³ + b³ Questions (15 Questions).
- New SSC topic test: SSC CHSL Algebra Mock Test 01: Identities and x + 1/x Questions (15 Questions).
- Related notes: mensuration formulas for SSC CGL, trigonometry formulas for SSC CGL, unit digit tricks for SSC CGL, percentage tricks for SSC CGL, coordinate geometry formulas for SSC CGL and surds and indices for SSC CGL.
- Exam details: SSC CGL 2026 notification.
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