Question

Solve for x,

X^2 + (ax/a+x)^2=3a^2

Answer 1

$x^2+(\frac{ax}{a+x})^2=3a^2\\\\\frac{x^2(a+x)^2+a^2x^2}{(a+x)^2}=3a^2\\\\x^4+2ax^3+2a^2x^2=3a^4+3a^2x^2+6a^3x\\\\x^4+2ax^3-a^2x^2-6a^3x-3a^4=0$

Divide the equation by a^4
let  x/a = y

Then:  y⁴ + 2 y³ - y² - 6 y - 3 =  0    --- (1)

We have to solve this equation for y.
That polynomial  can be written as :  (y² - a y - b) (y² + c y + 3/b) = 0   --- (2)
     expanding (2) and comparing with (1), we get:
             a = 1   ,  b = 1.    c = 3

So we get:    (y² - y - 1) ( y² + 3y + 3) = 0
             Solving  y² - y - 1 = 0, we get:     y = (1+√5)/2  or  (1-√5)/2

So  x = a(1+√5)/2 ,    or  a(1-√5)/2