Solving a simple rational equation $(\frac{6x}{6-x})^2+x^2=400$

Question

Clearly we could multiply both sides of $$\left(\dfrac{6x}{6-x}\right)^2+x^2=400$$ by $(6-x)^2$ which leads to a degree 4 polynomial equation, which we can solve using the bi-quadratic formula. Moreover, we could approximate the solutions using Newtons method. However, I have a feeling there is a much more graceful way to solve this. Any help is appreciated.

Answer 1

Please note that,you have:
$\left(\dfrac{6x}{6-x}\right)^2+x^2=400$.......(A)

Now,
$[\left(\dfrac{6x}{6-x}\right) -x]^2$=
$\left(\dfrac{x^2}{6-x}\right)^2$=$400$-$12$$\left(\dfrac{x^2}{6-x}\right)$...[using $(a-b)^2$ identity followed by(A)]
Now put $t=\left(\dfrac{x^2}{6-x}\right)$ [Keeping in mind that $x\ne6$]and you get a quadratic in $t$ which gives you two values of $t$.Then equate each $t$ with $\left(\dfrac{x^2}{6-x}\right)$,solve for $x$.....etc.
P.S.:In case of difficulty ,please comment.

Answer 2

The quartic (or bi-quadratic) polynomial for $x$ can be factored into two quadratic polynomials with coefficients in $\mathbb{Z}[\sqrt{109}]$, so that the closed-form expression for roots is not as terrible as it can be generally.

Clearing fractions one gets:

$$ x^4 - 12x^3 - 328x^2 + 4800x - 14400 = 0 $$

Then notice that:

$$ x^4 - 12x^3 - 328x^2 + 4800x - 14400 = (x^2 - Ax + 6A)(x^2 - Bx + 6B) $$

provided $A+B = 12$ and $AB = -400$.

It follows that the roots for $x$ may be found by solving the respective quadratic equations with $A = 6 + 2\sqrt{109}$ and $B = 6 - 2\sqrt{109}$.

More explicitly:

$$ x = 3 - \sqrt{109} \pm \sqrt{82 + 6\sqrt{109}}\;,\; 3 + \sqrt{109} \pm \sqrt{82 - 6\sqrt{109}} $$

A crude estimate shows that the expressions under the radicals are all positive, and thus the four roots are indeed real roots.

How was the reducibility of the polynomial recognized?

Initially I saw that setting $y = \frac{6x}{x-6}$ leads to a system of equations:

$$ x^2 + y^2 = 20^2 $$

$$ (x - 6)(y - 6) = 36 $$

Geometrically this asks for the intersection of a circle of radius $20$ centered at the origin and a "right" hyperbola with center $(6,6)$. Since both curves are symmetric about the line $y = x$, it follows that when $(x,y)$ is a point of intersection, so too is $(y,x)$.

Algebraically this amounts to saying that $x \mapsto \frac{6x}{x-6}$ is an involution which pairs up the roots $x$. A bit of scratchwork showed that the pairing of roots in this way results in factors $x^2 - Ax + 6A$ and $x^2 - Bx + 6B$ where $A,B$ are roots of $w^2 - 12w - 400 = 0$.