Difficulty: Easy
Correct Answer: 7
Explanation:
Introduction / Context:
This is a classic algebra identity question where you are given P^2 + 1/P^2 and asked to find P + 1/P. The objective is to recognise and apply the correct identity rather than attempting to solve a quadratic equation for P and then back substitute, which would be slower.
Given Data / Assumptions:
- P is a non zero real number, so division by P is allowed.
- P^2 + 1/P^2 = 47.
- We must find the value of P + 1/P.
Concept / Approach:
The key identity relating these expressions is (P + 1/P)^2 = P^2 + 2 + 1/P^2. Knowing P^2 + 1/P^2 allows you to compute (P + 1/P)^2 directly. After that, you take a square root, remembering that both positive and negative roots are possible in general, but aptitude questions typically expect the positive value unless otherwise specified.
Step-by-Step Solution:
Start from the identity (P + 1/P)^2 = P^2 + 2 + 1/P^2.
Substitute the given value P^2 + 1/P^2 = 47.
Then (P + 1/P)^2 = 47 + 2 = 49.
So P + 1/P is a real number whose square is 49.
Therefore P + 1/P = 7 or P + 1/P = -7.
Among the provided options, only 7 appears, so the intended answer is 7.
Verification / Alternative check:
If P + 1/P = 7, then P^2 + 1/P^2 is (P + 1/P)^2 - 2 = 7^2 - 2 = 49 - 2 = 47, which matches the given value. This confirms that choosing 7 is consistent with the original relation.
Why Other Options Are Wrong:
If you try P + 1/P = 5, then P^2 + 1/P^2 = 25 - 2 = 23, not 47. For 6 we get 36 - 2 = 34, for 8 we get 64 - 2 = 62, and for 4 we get 16 - 2 = 14. None of these match the required 47, so these options are incorrect.
Common Pitfalls:
Students sometimes mistakenly use P^2 + 1/P^2 = (P + 1/P)^2, forgetting the extra 2 term, which would lead to the wrong value. Others try to solve P^2 + 1/P^2 = 47 as a quadratic in P^2, which is unnecessary and complicated. Recognising the simple square identity is the most efficient approach.
Final Answer:
The value of the sum is 7.
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