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

If $\alpha, \beta, \gamma$ are the real roots of the equation $x^3 - 3px^2 + 3qx - 1 = 0$,then find the centroid of the triangle whose vertices are $(\alpha, \frac{1}{\alpha}), (\beta, \frac{1}{\beta})$ and $(\gamma, \frac{1}{\gamma})$.

The number of integers $k$ for which the equation $x^3-27x+k=0$ has at least two distinct integer roots is

If one root of the equation $ax^2 + bx + c = 0$ is the square of the other,then $a(c - b)^3 = cX$,where $X$ is

Which of the following quadratic equations has real roots $x_1, x_2$ that satisfy the conditions $x_1^2+x_2^2=5$ and $3(x_1^5+x_2^5)=11(x_1^3+x_2^3)$?

If $\alpha, \beta$ and $\gamma$ are the roots of the equation $x^{3}+4x+2=0$,then $\alpha^{3}+\beta^{3}+\gamma^{3}$ is equal to