Question

Two different positive numbers a and b each differ from their reciprocals by 1. What is a + b

Answer 1

Answer:

√5

Step-by-step explanation:

Each of the numbers a and b is a solution to |x-1/x|=1

Hence it is either a solution to x-1/x=1 or to (1/x)-x=1.

Then it must be a solution either to  x² - x - 1 =0 or x² + x - 1 =0

There are in total four such values of x, namely (±1 ±√5)/2

Out of these, two are positive: (-1+√5)/2  and (1+√5)/2. We can easily check that both of them indeed have the required property, and their sum is   (-1+√5)/2 + (1+√5)/2 = √5