The number of roots of the quadratic equation $8\sec^2 \theta - 6\sec \theta + 1 = 0$ is

Let $a, b, c$ be real numbers with $a \ne 0$. If $\alpha$ is a root of $a^2x^2 + bx + c = 0$,$\beta$ is a root of $a^2x^2 - bx - c = 0$,and $0 < \alpha < \beta$,then the equation $a^2x^2 + 2bx + 2c = 0$ has a root $\gamma$ that always satisfies:

If $\alpha, \beta, \gamma$ are roots of $x^3 - 2x^2 + 3x - 2 = 0$,then the value of $\left( \frac{\alpha\beta}{\alpha + \beta} + \frac{\alpha\gamma}{\alpha + \gamma} + \frac{\beta\gamma}{\beta + \gamma} \right)$ is

If $a^{2}=by+cz, b^{2}=cz+ax, c^{2}=ax+by,$ then the value of $\frac{x}{a+x}+\frac{y}{b+y}+\frac{z}{c+z}$ is

Find the value of $x^{3}+y^{3}+z^{3}-3xyz$,when $x+y+z=16$ and $xy+yz+zx=78$.

If $\alpha$ and $\beta$ are the roots of $ax^2 + bx + c = 0$ and $\alpha + \beta$,$\alpha^2 + \beta^2$,and $\alpha^3 + \beta^3$ are in $G.P.$,where $\Delta = b^2 - 4ac$,then: