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If the difference between a number and its reciprocal is 5, then what is the sum of the square of the number and its reciprocal?

If the difference between a number and its reciprocal is 5, then what is the sum of the square of the number and its reciprocal? Correct Answer 27

Given: 

The difference between a number and its reciprocal is 5

Concept used: 

(a - b)2 = a2 + b2 - 2ab

If b = 1/a, then

(a - 1/a)2 = a2 + 1/a2 - 2

or, a2 + 1/a2 = (a - 1/a)2 + 2

Calculation: 

Let the number be x and their reciprocal be 1/x

⇒ x - 1/x = 5

Now, Squaring both sides

⇒ (x - 1/x)2 = 52 

⇒ x2 + 1/x2 - 2 = 25

⇒ x2 + 1/x2 = 25 + 2

⇒ x2 + 1/x2 = 27

Hence, the required value is 27. 

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