If α and β are the roots of $x^2-8x+3=0$, the equation with integer coefficients whose roots are α/β and β/α is:

The new roots multiply to (α/β)(β/α) = 1 and add to (α² + β²)/(αβ) = $(64-6)/3=58/3$. So $x^2-(58/3)x+1=0$, and clearing the fraction gives $3x^2-58x+3=0$. (Using $(\alpha +\beta {)}^2=64$ without the −2αβ step gives the first option; forgetting to multiply the constant 1 by 3 gives the last one.) The roots are 4 ± √13 ≈ 7.6056 and 0.3944, whose ratios 19.284 and 0.0519 add to 19.336 ≈ 58/3 and multiply to 1.