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

1. Basic Identity

If:

x + 1/x = a

Then the following identities are useful:

x² + 1/x² = a² − 2

x³ + 1/x³ = a³ − 3a

These formulas are commonly used in algebra simplification problems.

2. When x − 1/x = a

If:

x − 1/x = a

Then:

x² + 1/x² = a² + 2

x³ − 1/x³ = a³ + 3a

These identities are also useful in competitive exam problems.

3. Fourth Power Identity

The following factorization identity is useful:

a⁴ + a²b² + b⁴ = (a² + ab + b²)(a² − ab + b²)

4. Fifth Power Relation

For expressions involving x + 1/x:

x⁵ + 1/x⁵ = (x² + 1/x²)(x³ + 1/x³) − (x + 1/x)

5. Special Values

Value	Result
x + 1/x = 2	x = 1
x + 1/x = −2	x = −1
x + 1/x = 1	x³ = −1
x + 1/x = −1	x³ = 1

6. General Identity

If:

ax + b/x = k

Then:

a²x² + b²/x² = k² − 2ab

a³x³ + b³/x³ = k³ − 3abk

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