Condition for fourth degree polynomial to have all real roots For what range of values of $a$ will the following fourth degree polynomial have all real roots:
$$x^4 - 2ax^2 + x + a^2 -a = 0$$

First transform this equation into the equation with unknown $a$:
$a^{2}-(2x^{2} + 1)a + x^{4} + x = 0$
Solving the equation with unknown obtain two second degree equations:
$x^{2} + x - a = 0 $ and $ x^{2} - x + 1 - a = 0$
For $a < - \frac{1}{4}$ equation has no real roots
For $a = - \frac{1}{4}$ equation has two equal real roots
For $- \frac{1}{4} < a < \frac{3}{4}$ equation has two different real roots
For $a=\frac{3}{4}$ equation has real four roots of which three are equal
For $ a> \frac{3}{4}$ equation has four different real roots
