Well, I was given a problem,
find $x$, if:
$$(x^2+100)^2=(x^3-100)^3$$
I tried everything that I could, I even opened up the brackets which gave an ugly degree 9 equation, I also tried to plot the curves $y=\left(x^2+100\right)^2$ and $y=\left(x^3-100\right)^3$ and locate their point of intersection but it couldn't be done manually.
So, in the end I was forced to use hit and trial after doing which I got the answer, is their any way to solve this algebraically??
Best Answer
There is a quite dirty trick you can use here. Define $$ f(x,t)=(x^2+t)^2-(x^3-t)^3\ . $$ Then solve $f(x,t)=0$ for $t$ (not for $x$). There is a simple solution of the cubic equation in $t$ $$ t=x^2(x-1)\ . $$ Then set $t=100$ and solve the cubic equation for $x$, yielding $x=5$ as the only real root.