The roots of the equation $3 a^{2} x^{2}-a b x-2 b^{2}=0$ are

The equation $\frac{3}{x - a^3} + \frac{5}{x - a^5} + \frac{7}{x - a^7} = 0$,where $a > 1$,has:

If $\alpha$ and $\beta$ are the roots of the equation $x^{2}+px+2=0$ and $\frac{1}{\alpha}$ and $\frac{1}{\beta}$ are the roots of the equation $2x^{2}+2qx+1=0$,then $\left(\alpha-\frac{1}{\alpha}\right)\left(\beta-\frac{1}{\beta}\right)\left(\alpha+\frac{1}{\beta}\right)\left(\beta+\frac{1}{\alpha}\right)$ is equal to:

For the quadratic equation $2x^2 - (p + 1)x + (p - 1) = 0$,if $\alpha - \beta = \alpha \beta$,then what is the value of $p$?

For what value of $k$ can the quadratic polynomial $3z^{2} + 5z + k$ be factored into a product of real linear factors?

Let $p(x)$ be a quadratic polynomial such that $p(0) = 1$. If $p(x)$ leaves a remainder of $4$ when divided by $x - 1$ and a remainder of $6$ when divided by $x + 1$,then: