If $\alpha, \beta$ are the roots of the quadratic equation $x^{2}-8x+k=0$,find the value of $k$ such that $\alpha^{2}+\beta^{2}=40$.

The two roots of the equation $x^3 - 9x^2 + 14x + 24 = 0$ are in the ratio $3 : 2$. The roots are:

If the sum of the squares of the roots of the equation $x^2 - (\sin \alpha - 2)x - (1 + \sin \alpha) = 0$ is least,then $\alpha$ is

In a cubic equation,the coefficient of $x^2$ is zero and the remaining coefficients are real. If one root is $\alpha = 3 + 4i$ and the remaining roots are $\beta$ and $\gamma$,then find the value of $\alpha \beta \gamma$.

Let $p, q$ and $r$ be real numbers $(p \ne q, r \ne 0),$ such that the roots of the equation $\frac{1}{x + p} + \frac{1}{x + q} = \frac{1}{r}$ are equal in magnitude but opposite in sign. Then the sum of squares of these roots is equal to:

The roots of the equation $x^2 + bx - c = 0$ where $b, c > 0$ are: