<s>[INST] Solve $x^2+x^2t^2+2xt^2+t^2=1$ for $x$. Solving
$$x^2+x^2t^2+2xt^2+t^2=1$$
for $x$ yields
$$x=\frac{1-t^2}{t^2+1}.$$
I had to use software because I could not manipulate it algebraically, and sadly the program did not tell me how it reached that solution.
What is the method?
 [/INST] Collect the coefficients in $x$ to get a polynomial in $x$:
$$ (t^2+1) x^2 +(2t^2)x+(t^2-1) = 0 $$
Now solve using quadratic formula:
$$ \frac{-2t^2 \pm \sqrt{(2t^2)^2 - 4(t^2+1)(t^2-1)}}{2(t^2+1)} = \\
 \frac{-2t^2 \pm 2}{2(t^2+1)} = \frac{-t^2 \pm 1}{t^2 + 1}$$
So either $x = -1$ or $x = \frac{-t^2 + 1}{t^2 + 1}$.
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