The roots of the equation $3^{2x} - 10 \cdot 3^x + 9 = 0$ are

Let $\alpha, \beta, \gamma$ be the three roots of the equation $x^3+bx+c=0$. If $\beta \gamma=1=-\alpha$,then $b^3+2c^3-3\alpha^3-6\beta^3-8\gamma^3$ is equal to $......$.

The value of $\lambda$ such that the sum of the squares of the roots of the quadratic equation $x^2 + (3 - \lambda)x + 2 = \lambda$ has the least value is

Find the maximum and minimum values of the function $f(x) = 9x^{2} + 12x + 2$.

Let $-\frac{\pi}{6} < \theta < -\frac{\pi}{12}$. Suppose $\alpha_1$ and $\beta_1$ are the roots of the equation $x^2 - 2x \sec \theta + 1 = 0$ and $\alpha_2$ and $\beta_2$ are the roots of the equation $x^2 + 2x \tan \theta - 1 = 0$. If $\alpha_1 > \beta_1$ and $\alpha_2 > \beta_2$,then $\alpha_1 + \beta_2$ equals

If the sum of two particular roots of the equation $x^4-4x^3-7x^2+22x+24=0$ is equal to the sum of the remaining two roots,then the sum of the cubes of all the roots of this equation is