Given \(x + 1/x = 5\) , determine the value of \(x^2 + 1/x^2\) .

Question

Given \(x + 1/x = 5\) , determine the value of \(x^2 + 1/x^2\) .

1. A. 25
2. B. 24
3. C. 26
4. D. 23 (Correct answer)

Correct Answer

Option D — 23

Detailed Solution & Explanation

The correct answer is 23.

Key Points

• Square both sides of the given equation, using (a + b)² = a² + 2ab + b²:
  • (x + 1/x)² = x² + 2·x·(1/x) + 1/x² = x² + 2 + 1/x²
• Since x + 1/x = 5, the left side is 25:
  • x² + 1/x² + 2 = 25 → x² + 1/x² = 23

Additional Information

• The middle term is always 2, because x × 1/x = 1 — that cancellation is what makes this family of identities work.
• The standard results worth memorising:
  • If x + 1/x = a, then x² + 1/x² = a² − 2 and x³ + 1/x³ = a³ − 3a
  • If x − 1/x = a, then x² + 1/x² = a² + 2
• Here x³ + 1/x³ would be 125 − 15 = 110, a frequent follow-up.
• Option A (25) is the trap for anyone who squares without subtracting the middle term.

प्रश्न (हिन्दी में)

दिया है \(x + 1/x = 5\) , तो \(x^2 + 1/x^2\) का मान ज्ञात करें।

1. A. 25
2. B. 24
3. C. 26
4. D. 23