<s>[INST] Suppose a is a real number for which all the roots of the equation $ x^4 - 2ax^2 + x +a^2 -a =0$ are real.Find the range of values of a? I have tried the question with $Rolle's$ theorem...
Let the equation have $2$ unequal real roots
$f'(x)= 4ax^3-4ax+1=0$
But cannot proceed after this.
 [/INST] we can write above equation as $x^4-2ax^2+x+a^2-a=0$ as $a^2-(2x^2+1)a+x^4+x=0$
So $$\displaystyle a= \frac{(2x^2+1)\pm\sqrt{(2x^2+1)^2-4(x^4+x)}}{2}=\frac{(2x^2+1)\pm (2x-1)}{2}$$
So $x^2+x-a=0$ or $x^2-x+1-a=0$
So for real roots , $1+4a\geq 0$ and  $1-4(1-a)\geq 0 $
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