If $\mathrm{\alpha },\mathrm{\beta }$ are the irrational roots of x⁵-5x⁴+9x³-9x²+5x-1=0 then the roots of equation $(\mathrm{\alpha }+\mathrm{\beta }){\mathrm{x}}^2-2\mathrm{\alpha \beta x}-\mathrm{\alpha \beta }=0$ are  

Roots of x⁵-5x⁴+9x³-9x²+5x-1=0 are $1,-\mathrm{\omega },-{\mathrm{\omega }}^2,\frac{3\pm \sqrt{5}}{2}$ . Irrational roots $\mathrm{\alpha }=\frac{3+\sqrt{5}}{2},\mathrm{\beta }=\frac{3-\sqrt{5}}{2}$ say then
$\mathrm{\alpha }+\mathrm{\beta }=3,\mathrm{\alpha \beta }=2$
$\because$ Required equation is 3x²+2x-1=0 roots are $-1,\frac{1}{3}$