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Quantitative Aptitude›Ch 4: Algebra, Geometry & Mensuration›Algebraic Identities
Master core formulas, 1/a patterns and value-putting tricks for competitive exams
Algebraic identities transform complex expansions into fast mental calculations. By mastering core formulas, 1/a reciprocal series and smart value-putting, you can bypass lengthy traditional algebra. Look for symmetry in equations to substitute values like 1 or 0 safely.
Memory Hooks
| Identity / Concept | Standard Formula | Shortcut / Result |
|---|---|---|
| Square Sum | (a+b)2 = a2+b2+2ab | (a-b)2 + 4ab |
| Reciprocal Square | x + 1/x = k | x2 + 1/x2 = k2 - 2 |
| Cube Sum | a3 + b3 = (a+b)(a2-ab+b2) | (a+b)3 - 3ab(a+b) |
| Reciprocal Cube | x + 1/x = k | x3 + 1/x3 = k3 - 3k |
(a+b)2
(Area expansion box)
Area expansion box
Visually divides a square into a2, b2 and two ab rectangles.
x + 1/x = k
(Reciprocal balance)
Reciprocal balance
Squaring both sides automatically eliminates middle terms.
Value Put
(Smart substitution)
Smart substitution
Put a=1, b=1 when equations have no denominator zero error.
a3 + b3
(Fast track formula)
Fast track formula
Use factored form to cancel matching terms in numerator and denominator.
x + 1/x = k implies x2 + 1/x2 = k2 + 2x2 + 1/x2 = k2 - 2
Plus sign on left changes to minus sign when squaring over to the right side.
Putting 0 for variables in fractional expressionsCheck denominator safety before putting 0
Never assign 0 if it creates an undefined division by zero error in options.
(a-b)2 = a2 + b2 - 2ab is same as (a+b)2Middle term sign differs (-2ab vs +2ab)
Be extremely careful with negative signs in binomial expansions.
Q1.If x + 1/x = 5, what is the value of x2 + 1/x2?
Q2.If a + b = 7 and ab = 12, what is the value of a3 + b3?