If $72^x \cdot 48^y = 6^{xy}$,where $x$ and $y$ are non-zero rational numbers,then $x+y$ equals

Let $a, b, c$ be three real numbers such that $a + 2b + 4c = 0$. Then the equation $ax^2 + bx + c = 0$

Let $S$ be the set of all non-zero real numbers $\alpha$ such that the quadratic equation $\alpha x^2 - x + \alpha = 0$ has two distinct real roots $x_1$ and $x_2$ satisfying the inequality $|x_1 - x_2| < 1$. Which of the following intervals is(are) a subset$(s)$ of $S$?
$(A) \left(-\frac{1}{2}, -\frac{1}{\sqrt{5}}\right)$
$(B) \left(-\frac{1}{\sqrt{5}}, 0\right)$
$(C) \left(0, \frac{1}{\sqrt{5}}\right)$
$(D) \left(\frac{1}{\sqrt{5}}, \frac{1}{2}\right)$

$4+\frac{1}{4+\frac{1}{4+\frac{1}{4+\ldots \infty}}} = $

The sum of all non-integer roots of the equation $x^5-6x^4+11x^3-5x^2-3x+2=0$ is

If ${x_1}, {x_2}, {x_3}$ are distinct roots of the equation $ax^2 + bx + c = 0$,then: