Question 1208661
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You start from

    {{{(x/a)}}} + {{{(x/b)^2}}} = 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

    {{{(x/b)^2}}} = c - {{{x/a}}}.


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

    {{{(x/b)^2}}} = {{{(ac - x)/a}}}.


It is the same as 

    {{{x^2/b^2}}} = {{{(ac-x)/a}}}.


Since  {{{b^2}}}  is now in the denominator, turn both fractions upside down

    {{{b^2/x^2}}} = {{{a/(ac-x)}}}.


Multiply both sides by  {{{x^2}}}

    {{{b^2}}} = {{{(ax^2)/(ac-x)}}}.


Now take square roots of both sides

    b = +/- {{{sqrt((ax^2)/(ac-x))}}} = +/- {{{abs(x)*sqrt(a/(ac-x))}}}.


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.
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Solved.