Question

If α ≠β but α²=5α-3, β²=5β-3, then find the equation whose roots are α/β and β/α

Solution

α²=Sα-3 → α²-sα+3=0

→ α = s±√2s-12/2= 5±√13/2

β^2='sβ -3 → β^2-sβ+3=0

→β = s±√2s-12/2=6±√13/2

since α≠β so, if α = 5-√13/2

β =5-√13/2

Sum of roots= α/β+β/α = α^2+β^2/αβ

=(α+β)^2-2αβ/αβ

=(5+√13/2+5-√13/2)^2-2(25-13/2)/25-13

25-12/12-13/12

Product of roots 6 α/β×β/α=1