Finding sum of reciprocals

Question

The sum of two positive numbers is 6, and the sum of their squares is 22. What is the sum of their reciprocals?

Answer 1

x+y=6

(x+y)²=x²+2xy+y²=36

x²+y² + 2xy = 36

22 + 2xy = 36

2xy = 14

xy = 7

1/x + 1/y = (y+x)/xy = 6/7

Answer 2

Given:

a+b=6

a² + b² = 22

so (a+b)² = 6² = 36

Also (a+b)² = a² + b² +2ab = 22 +2ab

equate both equations: 36 = 22 +2ab

solving: ab = 7

Find: 1/a + 1/b

Use common denominator to get (a+b)/ab, plug in values of (a+b) and ab to yield 6/7.