If $3$ distinct real numbers $a, b, c$ satisfy $a^2(a + p) = b^2(b + p) = c^2(c + p)$ where $p \in \mathbb{R}$,then the value of $bc + ca + ab$ is:

If the roots of the equation $x^2 + bx + ac = 0$ are $\alpha, \beta$ and the roots of the equation $x^2 + ax + bc = 0$ are $\alpha, \gamma$,then what are the values of $\alpha, \beta, \gamma$ respectively?

If $\tan \alpha$ and $\tan \beta$ are the roots of the equation $x^2 - px + q = 0$,then find the value of $\sin^2 (\alpha + \beta)$.

Let $\alpha$ and $\beta$ be two distinct roots of the equation $x^3 + 3x^2 - 1 = 0$. The equation which has $(\alpha \beta)$ as its root is equal to

If $\alpha, \beta$ are the roots of the equation $x^2 - 5x + 16 = 0$ and $(\alpha^2 + \beta^2)$ and $\frac{\alpha\beta}{2}$ are the roots of the equation $x^2 + px + q = 0$,then:

Let $p$ and $q$ be real numbers such that $p \neq 0, p^3 \neq q$ and $p^3 \neq -q$. If $\alpha$ and $\beta$ are non-zero numbers satisfying $\alpha + \beta = -p$ and $\alpha^3 + \beta^3 = q$,then find the quadratic equation whose roots are $\frac{\alpha}{\beta}$ and $\frac{\beta}{\alpha}$.