Algebra
SSC-CGL Exams
1. Overview
Algebra forms a key part of SSC CGL Quant — questions often test your grasp of identities, factorization, and equations through smart simplifications. With the right tricks, you can solve even tough algebraic problems in under 30 seconds.
2. Basic Algebraic Identities
Memorize and understand these — SSC repeatedly uses them for simplification:
| No. | Identity | Expansion |
|---|---|---|
| (1) | (a + b)² | a² + 2ab + b² |
| (2) | (a - b)² | a² - 2ab + b² |
| (3) | (a + b)(a - b) | a² - b² |
| (4) | (a + b + c)² | a² + b² + c² + 2(ab + bc + ca) |
| (5) | a³ + b³ | (a + b)(a² - ab + b²) |
| (6) | a³ - b³ | (a - b)(a² + ab + b²) |
| (7) | (a + b)³ | a³ + 3a²b + 3ab² + b³ |
| (8) | (a - b)³ | a³ - 3a²b + 3ab² - b³ |
Example 1
If a + b = 10 and ab = 21, find a² + b².
(a + b)² = a² + 2ab + b²
100 = a² + b² + 42 ⇒ a² + b² = 58
a² + b² = 58
Example 2
If a - b = 4 and ab = 45, find a² + b².
(a - b)² = a² + b² - 2ab ⇒ 16 = a² + b² - 90 ⇒ a² + b² = 106
a² + b² = 106
3. Factorization
Factorization means breaking algebraic expressions into simpler products.
| Type | Formula |
|---|---|
| Difference of squares | a² - b² = (a + b)(a - b) |
| Perfect square trinomials | x² + 2xy + y² = (x + y)² |
| Cubic forms | a³ ± b³ = (a ± b)(a² ∓ ab + b²) |
| Common factor | Take out common term |
| Middle term splitting | For quadratics like ax² + bx + c |
Example 3
Factorize x² + 5x + 6
→ Split middle term: 2 + 3 = 5
x² + 2x + 3x + 6 = (x + 2)(x + 3)
(x + 2)(x + 3)
Example 4
Factorize 9x² - 25y²
= (3x + 5y)(3x - 5y)
(3x + 5y)(3x - 5y)
4. Simplification
Algebraic simplification involves reducing using identities, cancellation, and substitution.
Example 5
If x + 1/x = 3, find x² + 1/x².
(x + 1/x)² = x² + 1/x² + 2 ⇒ 9 = x² + 1/x² + 2 ⇒ x² + 1/x² = 7
7
Example 6
If x + 1/x = 4, find x³ + 1/x³.
(x + 1/x)³ = x³ + 1/x³ + 3(x + 1/x)
64 = x³ + 1/x³ + 12 ⇒ x³ + 1/x³ = 52
52
5. Linear Equations
A linear equation is of the form:
ax + b = 0
Solution: x = -b/a
Example 7
Solve: 5x - 10 = 0
x = 10/5 = 2
x = 2
Example 8
Solve the pair: 2x + 3y = 12 and 3x - 2y = 5
Multiply first by 2 and second by 3 to eliminate y:
4x + 6y = 24
9x - 6y = 15
Add ⇒ 13x = 39 ⇒ x = 3
Substitute ⇒ 2(3) + 3y = 12 ⇒ y = 2
x = 3, y = 2
6. Quadratic Equations
Quadratic Equation:
ax² + bx + c = 0
x = [-b ± √(b² - 4ac)] / 2a
Example 9
Solve: x² - 5x + 6 = 0
x = [5 ± √(25 - 24)] / 2 = (5 ± 1)/2
x = 3 or x = 2
Example 10
Solve: 2x² - 7x + 3 = 0
x = [7 ± √(49 - 24)] / 4 = (7 ± 5)/4
x = 3 or ½
7. Short Tricks for SSC
| Type | Shortcut | Example |
|---|---|---|
| x + 1/x = a | x² + 1/x² = a² - 2 | If 3 → 7 |
| x + 1/x = a | x³ + 1/x³ = a³ - 3a | If 4 → 52 |
| a² + b² | (a + b)² - 2ab | - |
| a³ + b³ | (a + b)³ - 3ab(a + b) | - |
| Two roots of eqn | Product = c/a, Sum = -b/a | x² - 5x + 6 = 0 ⇒ 2,3 |
| Discriminant | b² - 4ac > 0 → real roots | Used to check type of roots |
8. SSC-Level Examples
Example 11
If a + b = 7 and ab = 10, find a³ + b³.
(a + b)³ - 3ab(a + b) = 343 - 210 = 133
133
Example 12
If x + 1/x = 5, find x⁴ + 1/x⁴.
(x² + 1/x²) = 25 - 2 = 23
x⁴ + 1/x⁴ = 23² - 2 = 527
527
Example 13
If x - 1/x = 2, find x² + 1/x².
(x - 1/x)² = x² + 1/x² - 2 ⇒ 4 = x² + 1/x² - 2 ⇒ 6
6
9. Practice Section
Q1. If a + b = 9 and ab = 20, find a² + b².
View Answer
(a + b)² = a² + 2ab + b² ⇒ 81 = a² + b² + 40 ⇒ a² + b² = 41
41
Q2. If x + 1/x = 6, find x³ + 1/x³.
View Answer
= 6³ − 3×6 = 216 − 18 = 198
198
Q3. Solve: x² - 11x + 24 = 0
View Answer
x = [11 ± √(121 − 96)] / 2 = (11 ± 5)/2 ⇒ 8, 3
x = 8 or 3
Q4. Simplify: (a² - b²)/(a - b)
View Answer
= (a + b)(a - b)/(a - b) = a + b
a + b
Q5. If x + 1/x = 2, find x² + 1/x².
View Answer
= 2² − 2 = 2
2
Q6. Find sum & product of roots of 2x² - 5x + 3 = 0.
View Answer
Sum = −b/a = 5/2, Product = c/a = 3/2
Sum = 5/2, Product = 3/2
Q7. If a + b = 12 and ab = 35, find a³ + b³.
View Answer
(a + b)³ − 3ab(a + b) = 1728 − 1260 = 468
468
10. Quick Recap Table
| Concept | Formula | Tip |
|---|---|---|
| a² + b² | (a + b)² − 2ab | Most-used |
| a³ + b³ | (a + b)³ − 3ab(a + b) | Remember signs |
| x + 1/x = a | x² + 1/x² = a² − 2 | Core identity |
| Quadratic roots | x = [-b ± √(b² − 4ac)] / 2a | Always simplify |
| Product & Sum | P = c/a, S = -b/a | Helps check |
| Factorization | Take out common term or use identities | Save time |
You've completed Article 9: Algebra!
Courage Tip: Focus on pattern recognition — SSC questions often hide a simple identity behind a complex-looking expression. Spot the identity → apply → simplify fast.
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