Algebra (Basic Identities & Linear Equations) — SSC Quant Notes Ch. 18

Introduction

SSC's Algebra section rarely asks for lengthy derivations — instead, it tests whether you can instantly recognise and apply a small set of standard identities to simplify expressions and find values quickly.

Standard Algebraic Identities — Memorise These

Identity
(a+b)² = a² + 2ab + b²
(a−b)² = a² − 2ab + b²
a² − b² = (a+b)(a−b)
(a+b)² − (a−b)² = 4ab
(a+b)² + (a−b)² = 2(a²+b²)
a³ + b³ = (a+b)(a² − ab + b²)
a³ − b³ = (a−b)(a² + ab + b²)
(a+b)³ = a³ + b³ + 3ab(a+b)
(a−b)³ = a³ − b³ − 3ab(a−b)

The "a + 1/a" Family — Extremely High-Yield

Given	Find	Formula
a + 1/a = k	a² + 1/a²	k² − 2
a − 1/a = k	a² + 1/a²	k² + 2
a + 1/a = k	a³ + 1/a³	k³ − 3k
a − 1/a = k	a³ − 1/a³	k³ + 3k

Q. If a + 1/a = 5, find the value of a² + 1/a².

a² + 1/a² = k² − 2 = 5² − 2 = 25 − 2 = 23

Q. If x − 1/x = 3, find the value of x³ − 1/x³.

x³ − 1/x³ = k³ + 3k = 3³ + 3×3 = 27+9 = 36

Solving Linear Equations

Type	Method
One variable (ax + b = c)	Isolate x: x = (c−b)/a
Two variables (simultaneous equations)	Substitution or Elimination method

Q. Solve: 2x + 3y = 12 and x − y = 1.

From the 2nd equation: x = y+1
Substitute: 2(y+1) + 3y = 12 → 2y+2+3y = 12 → 5y = 10 → y = 2
x = y+1 = x = 3, y = 2

Common SSC Pattern — "Sum and Difference Given"

Q. The sum of two numbers is 25 and their difference is 5. Find the numbers.

Larger = (Sum+Difference)/2 = (25+5)/2 = 15
Smaller = (Sum−Difference)/2 = (25−5)/2 = 10 (numbers: 15 and 10)

💡 Exam Tip: The "a+1/a" family of formulas is one of the highest-frequency SSC Algebra patterns — memorising the four formulas above (rather than deriving them each time by squaring/cubing) can save 30-40 seconds per question.

✅ Practice Focus: Nine standard identities (squares, cubes, difference of squares) · The "a+1/a" / "a−1/a" formula family · Simultaneous linear equations (substitution/elimination) · Sum-and-difference shortcut for two-number problems.

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