If A and G be A.M. and GM., respectively between two positive numbers, prove that the numbers are `A+-sqrt((A+G)(A-G))`.

Explanation

Let two positive numbers be a and b
`AM=A=(a+b)/2`-----(1)
`GM=G=sqrtab`--------(2)
From (1) and (2) , `a+b=2A` ------(3)
`GM=G^2`----------(4)
Substituting the values of (3 ) and (4) in the identity `(a-b)^2=(a+b)^2-4ab `
`(a-b)^2=4A^2-4G^2`
`implies 4(A^2-G^2) `
`(a-b)^2=4(A+G)(A-G) `
`(a-b) =2 sqrt[(A+G)(A-G)]`------------(5)
From (3) and (5)
`2a=2A+2sqrt[(A+G)(A-G)]`
`a=A+sqrt[(A-G)(A+G)]`
Substituting the value of a in (3)
`b=2A=A-sqrt[(A-G)(A+G)`
`=A-sqrt[(A-G)(A+G)]`
Thus the two numbers are `A pm sqrt[(A-G)(A+G)]`