The roots of the quadratic equation $3x^2 - 5x - 1 = 0$ are $\alpha$ and $\beta$. Without solving the equation, find the exact value of $\alpha^2 + \beta^2$.

1. 由韦达定理求出两根之和与两根之积

对于二次方程 $3x^2 - 5x - 1 = 0$，根据根与系数的关系（韦达定理）：
$\alpha + \beta = -\frac{-5}{3} = \frac{5}{3}$
$\alpha\beta = \frac{-1}{3} = -\frac{1}{3}$

2. 变形代数式并代入求出 $\alpha^2 + \beta^2$ 的精确值

将 $\alpha^2 + \beta^2$ 用 $\alpha + \beta$ 和 $\alpha\beta$ 表示：
$\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta$
代入上述结果：
$\alpha^2 + \beta^2 = \left(\frac{5}{3}\right)^2 - 2\left(-\frac{1}{3}\right) = \frac{25}{9} + \frac{2}{3} = \frac{31}{9}$