If \(x = √5 + 2\) , then find the value of \(x^2 - 4√5\) .

Question

If \(x = √5 + 2\) , then find the value of \(x^2 - 4√5\) .

1. A. 4
2. B. 5
3. C. 9 (Correct answer)
4. D. 11

Correct Answer

Option C — 9

Detailed Solution & Explanation

The correct answer is 9.

Key Points

• Expand x² using (a + b)² = a² + 2ab + b² with a = √5 and b = 2:
  • x² = (√5)² + 2(√5)(2) + 2² = 5 + 4√5 + 4 = 9 + 4√5
• Subtract as asked: x² − 4√5 = (9 + 4√5) − 4√5 = 9.

Additional Information

• The surd term is designed to cancel, which is the signal that the question wants the expansion rather than a decimal approximation.
• Related identities that appear constantly: (a − b)² = a² − 2ab + b² and (a + b)(a − b) = a² − b².
• A useful companion result: if x = √5 + 2, then 1/x = √5 − 2, since their product is 5 − 4 = 1 — which makes x + 1/x = 2√5 and x − 1/x = 4.
• Questions of this shape are built so the irrational part cancels, so expand first and simplify before reaching for a calculator.
• The conjugate √5 − 2 is worth noting: (√5 + 2)(√5 − 2) = 5 − 4 = 1, which makes the two numbers exact reciprocals and underlies several follow-up identities.

प्रश्न (हिन्दी में)

यदि \(x = √5 + 2\) हो, तो \(x^2 - 4√5\) का मान ज्ञात कीजिए।

1. A. 4
2. B. 5
3. C. 9
4. D. 11