If α and β are the roots of $3x^2+5x-2=0$, then α/β + β/α equals:

α + β = −5/3 and αβ = −2/3. Now α/β + β/α = (α² + β²)/(αβ), and α² + β² = $(\alpha +\beta {)}^2-2$αβ = 25/9 + 4/3 = 37/9. So the value is $(37/9)÷(-2/3)=-37/6$. (Skipping the −2αβ step gives −25/6, and 37/9 is only the numerator.) Check with the actual roots 1/3 and −2: $(1/3)/(-2)+(-2)/(1/3)=-1/6-6=-37/6$.