If the roots of the equation $ax^2 + x + b = 0$ are real and distinct,then what will be the nature of the roots of the equation $x^2 - 4\sqrt{ab}x + 1 = 0$?

The quadratic equation with real coefficients whose one root is $7 + 5i$ will be:

If one root of the equation $x^2 + px + 12 = 0$ is $4$,and the equation $x^2 + px + q = 0$ has equal roots,then what is the value of $q$?

If $P(x) = ax^2 + bx + c$ and $Q(x) = -ax^2 + dx + c$ where $ac \neq 0$,then $P(x) \cdot Q(x) = 0$ has at least:

If $\alpha, \beta, \gamma$ are the roots of the equation $x^3+4x^2-9x-36=0$ and $\alpha < \beta < \gamma$,then find the value of $\alpha+2\beta+3\gamma$.

Let $-\frac{\pi}{6} < \theta < -\frac{\pi}{12}$. Suppose $\alpha_1$ and $\beta_1$ are the roots of the equation $x^2 - 2x \sec \theta + 1 = 0$ and $\alpha_2$ and $\beta_2$ are the roots of the equation $x^2 + 2x \tan \theta - 1 = 0$. If $\alpha_1 > \beta_1$ and $\alpha_2 > \beta_2$,then $\alpha_1 + \beta_2$ equals