Algebraic Identities for Important Formulas for Exams

In many competitive exams like SSC, Banking, and Railway, expressions involving $x + \frac{1}{x}$ frequently appear. These identities help simplify such expressions quickly.

1. Basic Identity

$x + \frac{1}{x} = a$

then the following identities are useful:

x^2 + \frac{1}{x^2} = a^2 – 2

x^3 + \frac{1}{x^3} = a^3 – 3a

These formulas are commonly used in algebra simplification problems.

2. When $x – \frac{1}{x} = a$

x – \frac{1}{x} = a

then

x^2 + \frac{1}{x^2} = a^2 + 2

x^3 – \frac{1}{x^3} = a^3 + 3a

These identities are also useful in competitive exam problems.

3. Fourth Power Identity

The following factorization identity is useful:

a^4 + a^2 b^2 + b^4 = (a^2 + ab + b^2)(a^2 – ab + b^2)

4. Fifth Power Relation

For expressions involving $x + \frac{1}{x}$ :

x^5 + \frac{1}{x^5} = \left(x^2 + \frac{1}{x^2}\right) \left(x^3 + \frac{1}{x^3}\right) – \left(x + \frac{1}{x}\right)

5. Special Values

Expression	Result
$x + \frac{1}{x} = 2$	$x = 1$
$x + \frac{1}{x} = -2$	x = -1
$x + \frac{1}{x} = 1$	$x ^ 3 = -1$
$x + \frac{1}{x} = -1$	$x^3 = 1$

6. General Identity

ax + \frac{b}{x} = k

then

a^2x^2 + \frac{b^2}{x^2} = k^2 – 2ab

a^3x^3 + \frac{b^3}{x^3} = k^3 – 3abk

Practice More

To practice algebra questions and improve speed, try mock tests on the Chitwan platform.