Find the nature of roots for quadratic equations and calculate expressions involving roots.

For the quadratic equation $ax^2 + bx + c = 0$, we have $a = 2$, $b = -5$, $c = 2$.

Since $\Delta = 9 > 0$, the roots are real and distinct.
The roots are also rational because the discriminant is a perfect square.
The nature of the roots is real and distinct.

For the quadratic equation $ax^2 + bx + c = 0$, we have $a = 3$, $b = -7$, $c = 4$.

Since $\Delta = 1 > 0$, the roots are real and distinct.
The roots are also rational because the discriminant is a perfect square.
The nature of the roots is real and distinct.

If $\alpha$ and $\beta$ are the roots of the equation $3x^2 - 5x + 4 = 0$, find the value of $\frac{\alpha}{\beta} + \frac{\beta}{\alpha}$.

For the quadratic equation $ax^2 + bx + c = 0$, we have $a = 3$, $b = -5$, $c = 4$.

Sum of roots: $\alpha + \beta = -\frac{b}{a}$

Product of roots: $\alpha \beta = \frac{c}{a}$

We know that $\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha \beta$.
So, the expression becomes:

To subtract the terms in the numerator, find a common denominator:

To divide by a fraction, multiply by its reciprocal:

The value of $\frac{\alpha}{\beta} + \frac{\beta}{\alpha}$ is $\boxed{\frac{1}{12}}$.