What is the value of (2+√5)(2−√5)?
Answer and explanation
Correct answer: −1
The two factors are conjugates, so the difference-of-squares identity (a+b)(a−b) = a²−b² applies directly. Here a = 2 and b = √5. Thus (2+√5)(2−√5) = 2² − (√5)² = 4 − 5 = −1, making option B correct. Direct expansion confirms the result: 4 − 2√5 + 2√5 − 5. The middle terms cancel because they have equal magnitudes and opposite signs, leaving −1. Option A could arise from using the wrong sign between the squares. Option C ignores the fact that 2² is 4, and option D fails to simplify the conjugate product. The cancellation of the radical term is the key structural idea.
Frequently asked questions
What is the correct answer to this question?
−1
Why is this the correct answer?
The two factors are conjugates, so the difference-of-squares identity (a+b)(a−b) = a²−b² applies directly. Here a = 2 and b = √5. Thus (2+√5)(2−√5) = 2² − (√5)² = 4 − 5 = −1, making option B correct. Direct expansion confirms the result: 4 − 2√5 + 2√5 − 5. The middle terms cancel because they have equal magnitudes and opposite signs, leaving −1. Option A could arise from using the wrong sign between the squares. Option C ignores the fact that 2² is 4, and option D fails to simplify the conjugate product. The cancellation of the radical term is the key structural idea.
Which subject and chapter does this question cover?
This is a Class 9 Mathematics question. Chapter: Number Systems. Topic: Real Numbers.
Student feedback
Was this question useful?
👍 0 Helpful 👎 0 Not helpful
Yes 0% No 0%
0 responsesStudent Reviews
No published reviews yet.
