The sum of the solutions of the equation $|\sqrt{x} - 2| + \sqrt{x}(\sqrt{x} - 4) + 2 = 0$ for $x > 0$ is equal to

If $\alpha$ and $\beta$ are two complex roots of the equation $6x^6-25x^5+31x^4-31x^2+25x-6=0$,then $\alpha+\beta=$

The number of real roots of the equation $\sqrt{x^2-4x+3}+\sqrt{x^2-9}=\sqrt{4x^2-14x+6}$ is:

The expression $y = ax^2 + bx + c$ always has the same sign as $c$ if

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

The equation whose roots are squares of the roots of $x^4-2 x^3+6 x-21=0$ is