SOLUTION: Solve for b. (x/a) + (x/b)^2 = c

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Question 1208661: Solve for b.

(x/a) + (x/b)^2 = c
Answer by ikleyn(52781) About Me  (Show Source):
You can put this solution on YOUR website!
.


You start from

    %28x%2Fa%29 + %28x%2Fb%29%5E2 = c.


    As your starting equation is written in this form, 
    it is assumed implicitly that a =/=0,  b =/= 0.


Isolate the term with b in the left side

    %28x%2Fb%29%5E2 = c - x%2Fa.


Since left side is the fraction, write right side as a fraction, too

    %28x%2Fb%29%5E2 = %28ac+-+x%29%2Fa.


It is the same as 

    x%5E2%2Fb%5E2 = %28ac-x%29%2Fa.


Since  b%5E2  is now in the denominator, turn both fractions upside down

    b%5E2%2Fx%5E2 = a%2F%28ac-x%29.


Multiply both sides by  x%5E2

    b%5E2 = %28ax%5E2%29%2F%28ac-x%29.


Now take square roots of both sides

    b = +/- sqrt%28%28ax%5E2%29%2F%28ac-x%29%29 = +/- abs%28x%29%2Asqrt%28a%2F%28ac-x%29%29.


Any of these two final expressions is the desired expression for b.

This final expression is valid under assumption that  ac-x =/= 0 and the expression under the square root is positive.

Solved.