Given that $\alpha$ and $\beta$ are the roots of the quadratic equation $3x^2 - 5x + 1 = 0$, we need to find the value of $\alpha^2 - \alpha\beta + \beta^2$.

1. Problem Description

Given that $\alpha$ and $\beta$ are the roots of the quadratic equation $3x^2 - 5x + 1 = 0$, we need to find the value of $\alpha^2 - \alpha\beta + \beta^2$.

2. Solution Steps

First, we can rewrite the expression $\alpha^2 - \alpha\beta + \beta^2$ as $(\alpha^2 + \beta^2) - \alpha\beta$.

We know that $(\alpha + \beta)^2 = \alpha^2 + 2\alpha\beta + \beta^2$, which can be rearranged to $\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta$.

Substituting this into our target expression, we get:

$\alpha^2 - \alpha\beta + \beta^2 = (\alpha^2 + \beta^2) - \alpha\beta = ((\alpha + \beta)^2 - 2\alpha\beta) - \alpha\beta = (\alpha + \beta)^2 - 3\alpha\beta$.

For a quadratic equation of the form $ax^2 + bx + c = 0$, the sum of the roots is given by $\alpha + \beta = -\frac{b}{a}$ and the product of the roots is given by $\alpha\beta = \frac{c}{a}$.

In our case, the equation is $3x^2 - 5x + 1 = 0$, so $a = 3$, $b = -5$, and $c = 1$.

Therefore, $\alpha + \beta = -\frac{-5}{3} = \frac{5}{3}$ and $\alpha\beta = \frac{1}{3}$.

Now, we can substitute these values into the expression $(\alpha + \beta)^2 - 3\alpha\beta$:

$(\frac{5}{3})^2 - 3(\frac{1}{3}) = \frac{25}{9} - 1 = \frac{25}{9} - \frac{9}{9} = \frac{16}{9}$.

3. Final Answer

The value of $\alpha^2 - \alpha\beta + \beta^2$ is $\frac{16}{9}$.