Given quadratic: 3x² - 5x + a = 0 with roots α, β. Using sum and product formulas: α + β = 5/3, αβ = a/3.

The inequality simplifies to (α² + β²)/(αβ) > 2. Since α² + β² = (α+β)² - 2αβ, we get:

$\frac{{(\alpha +\beta )}^2-2\mathrm{\alpha \beta }}{\mathrm{\alpha \beta }}>2$

Substitute values: $\frac{{\left(\frac{5}{3}\right)}^2-2·\frac{a}{3}}{\frac{a}{3}}>2$

Solving gives: (25/9 - 2a/3)/(a/3) > 2 → (25 - 6a)/3a > 2 → 25 - 6a > 6a → 25 > 12a → a < 25/12

Since roots are real, discriminant ≥ 0: 25 - 12a ≥ 0 → a ≤ 25/12. Combining with a < 25/12 gives 0 < a < 25/12.

Final answer: $0<a<\frac{25}{12}$