If $\frac{1 + \sqrt{3}i}{2}$ is a root of the equation $x^4 - x^3 + x - 1 = 0$,then its real roots are:

The roots of $|x - 2|^2 + |x - 2| - 6 = 0$ are

What is the nature of the roots of the equation $a^2x^2 + (a + b)x - b^2 = 0$?

For $n > 2$ and $n \in N$,the product of the roots of $(x-n)((x^2-2nx)^2 + (2n^2-5)(x^2-2nx) + (n^4-5n^2+4)) = 0$ is divisible by

If $\alpha$ and $\beta$ are the real roots of the equation $\sqrt{\frac{5x}{x-2}} + \sqrt{\frac{x-2}{5x}} = \frac{29}{10}$ and $\alpha > \beta$,then $\sqrt{\alpha^2 - 11^4 \beta^2} = $

Let $\alpha$ and $\beta$ be the roots of the quadratic equation $x^2 \sin \theta - x(\sin \theta \cos \theta + 1) + \cos \theta = 0$ where $0 < \theta < 45^\circ$,and $\alpha < \beta$. Then $\sum_{n=0}^\infty (\alpha^n + \frac{(-1)^n}{\beta^n})$ is equal to