If the roots of the quadratic equation $x^2 - 4x - \log_3 a = 0$ are real,then what is the minimum value of $a$?

Let $f(x) = x^2 + ax + b$,where $a, b \in R$. If $f(x) = 0$ has all its roots imaginary,then the roots of $f(x) + f'(x) + f''(x) = 0$ are

If $(x-2)$ is a common factor of the expressions $x^2+ax+b$ and $x^2+cx+d$,then $\frac{b-d}{c-a}$ is equal to

If the roots of the equation $(p-3)x^2 + 2(p-3)x + 2p-5 = 0$ are real and distinct for $\alpha < p < \beta$ and $(\beta - \alpha)$ is maximum,then the extreme value of the quadratic expression $-(\alpha + \beta)x^2 + \alpha \beta x + (\alpha - \beta)$ is

For what set of values of $p$ do the roots of the equation $3x^2 + 2x + p(p - 1) = 0$ have opposite signs?

Solve the equation $\sqrt{3} x^{2}-\sqrt{2} x+3 \sqrt{3}=0$.