<s>[INST] can have solution of $x^4-3x^3+2x^2-3x+1=0$ using only high school methods can have solution of $x^4-3x^3+2x^2-3x+1=0$ using only high school methods???
i only know quadratic formula $$x =  \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ i tried many algebraic manipulations and i get $(x^2+1)^2=3(x^3+x)$, so can we have solution using only high school methods? i guess i have to do an algebraic substitution to reduce $x^3+x$ to polynomial of degree 2?? i also know $$(r_1 - r_2)^2 = (r_1 + r_2)^2 - 4r_1r_2$$ where $r_1,r_2$ are roots of polynomial of degree 2
 [/INST] we have $x^4-3x^3+2x^2-3x+1=0$ and we see that $x=0$ is not a solution, since we can write
$x^2+1/x^2-3(x+1/x)+2=0$
setting $t=x+1/x$ we get $x^2+1/x^2=t^2-2$ and we have a quadratic equation
$t^2-3t=0$
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