Solve $x^4+1-3(x^3+x)=2x^2$ analytically

Question

Solving $x^4+1-3(x^3+x)=2x^2$ is given as an exercise in Higher Algebra by Hall and Knight. Rearranging for $(x^2-1)^2=3x(x^2+1)$ we can sketch the graph to show there is a root in $(0,1)$ and since $x^4$ outgrows $x^3$ deduce there is a greater root

As useful as this form of the equation looks, it now seems to me it leads nowhere. How can I analytically solve this equation? I am more interested in solutions specific to this form of quartic such as a factorisation or algebra tricks I do not see(as the author would have intended), and less interested in solutions to the general quartic or similar approaches.

Divide all the thing by $x^2$ and let $t=x+\frac1x$, then $x^2+\frac1{x^2}=t^2-2$ and you'll left with $t^2-2-3t=2$. — Alexey Burdin, Aug 05 '20 at 09:06
See https://math.stackexchange.com/questions/480102/quadratic-substitution-question-applying-substitution-p-x-frac1x-to-2x4x — lab bhattacharjee, Aug 05 '20 at 09:17

score 2 · Answer 1 · answered Aug 05 '20 at 09:02

2

The equation is equivalent to $$ (x^2 + x + 1)(x^2 - 4x + 1)=0, $$ which is an algebraic solution, but I think still useful.

answered Aug 05 '20 at 09:02

Dietrich Burde

130,978

Solve $x^4+1-3(x^3+x)=2x^2$ analytically

1 Answers1