For a Given Number, Solve the Expression...

From: Anonymous
Date: Fri, 28 Jun 1996 9:22:58 -0400 (EDT) 
Subject: Solve the expression

I don't understand this problem. Can you help me?

For a given number m, m greater than or equal to 1 , solve: 
The square root of m + x plus the square root of m - x 
divided by the square root of m.

Date: Fri, 28 Jun 1996 13:33:35 -0400 (EDT)
From: Dr. Anthony
Subject: Re: Solve the expression

Start by multiplying through by the denominator to get: 

sqrt(m+x) + sqrt(m-x) = sqrt(m){sqrt(m+x) - sqrt(m-x)} 

Now square both sides using (a+b)^2 = a^2 + 2ab + b^2 to get after 
collecting terms:

2m + 2sqrt(m+x)*sqrt(m-x) = m{2m - 2sqrt(m+x)*sqrt(m-x)} 
cancel 2's and collect the sqrt(m+x)*sqrt(m-x) to one side: 

sqrt(m+x)*sqrt(m-x){1+m} = m^2 - m
sqrt(m+x)*sqrt(m-x) = m(m-1)/(m+1) 

Now square both sides again 

(m+x)(m-x) = m^2(m-1)^2/(m+1)^2

m^2 - x^2 = m^2(m-1)^2/(m+1)^2

x^2 = m^2 - m^2(m-1)^2/(m+1)^2

= {m^2(m+1)^2 - m^2(m-1)^2}/(m+1)^2

so x^2 = 4m^3/(m+1)^2

and x = +or- 2m*sqrt(m)/(m+1)

-Doctor Anthony, The Math Forum