Question

if m:n is the duplicate ratio of m+x :n+x show that x^2=mn

Answer 1

using the definition of duplicate ratio, m/n is equal to the square of (m+x)/(n+x). In other words 

m/n = ((m+x)/(n+x))^2 

isolate the term x^2 to get... 

m/n = ((m+x)/(n+x))^2
m/n = ((m+x)^2)/((n+x)^2)
m(n+x)^2 = n(m+x)^2
m(n^2+nx+x^2) = n(m^2+mx+x^2)
mn^2+mnx+mx^2 = nm^2+mnx+nx^2
mn^2+mx^2 = nm^2+nx^2
mx^2-nx^2 = nm^2-mn^2
(m-n)x^2 = nm^2-mn^2
x^2 = (nm^2-mn^2)/(m-n)
x^2 = (mn(m-n))/(m-n)
x^2 = mn