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
Practice More
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