Let $$\alpha, \beta$$ be the roots of the equation $$x^2+2 \sqrt{2} x-1=0$$. The quadratic equation, whose roots are $$\alpha^4+\beta^4$$ and $$\frac{1}{10}(\alpha^6+\beta^6)$$, is:

The sum of all the solutions of the equation $$(8)^{2 x}-16 \cdot(8)^x+48=0$$ is :

Let $$\alpha, \beta$$ be the distinct roots of the equation $$x^2-\left(t^2-5 t+6\right) x+1=0, t \in \mathbb{R}$$ and $$a_n=\alpha^n+\beta^n$$. Then the minimum value of $$\frac{a_{2023}+a_{2025}}{a_{2024}}$$ is

The coefficients $$\mathrm{a}, \mathrm{b}, \mathrm{c}$$ in the quadratic equation $$\mathrm{a} x^2+\mathrm{bx}+\mathrm{c}=0$$ are from the set $$\{1,2,3,4,5,6\}$$. If the probability of this equation having one real root bigger than the other is p, then 216p equals :