The roots of $|x - 2{|^2} + |x - 2| - 6 = 0$are

Let the sum of the maximum and the minimum values of the function $f(x)=\frac{2 x^2-3 x+8}{2 x^2+3 x+8}$ be $\frac{m}{n}$, where $\operatorname{gcd}(\mathrm{m}, \mathrm{n})=1$. Then $\mathrm{m}+\mathrm{n}$ is equal to :

Number of solutions of equation $|x^2 -2|x||$ = $2^x$ , is

Let $x_1, x_2, \ldots, x_6$ be the roots of the polynomial equation $x^6+2 x^5+4 x^4+8 x^3+16 x^2+32 x+64=0$. Then,

The number of ordered pairs $(x, y)$ of positive integers satisfying $2^x+3^y=5^{x y}$ is

The number of real solutions of the equation $|x{|^2}$-$3|x| + 2 = 0$ are