What are the conditions on $a$ such that the polynomial $x^4-2ax^2+x+a^2-a$ has four distinct real roots?

Note that the polynomial factors as $$(x^2+x-a)(x^2-x+1-a)$$ The discriminants are both positive if and only if $a\gt 3/4$. And if $a\gt 3/4$, subtraction shows the two quadratics cannot have a common root. Well within high school range.


The given equation can be written as $a^2-(2x^2+1)a+x^4+x=0$

$a=\frac{2x^2+1\pm\sqrt{(2x^2+1)^2-4(x^4+x)}}{2}=\frac{2x^2+1\pm(2x-1)}{2}$

$a=x^2+x$ or $a=x^2-x+1$

which are now quadratics in $x$

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Polynomials

Roots

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