Find a quadratic equation whose roots `x_(1)` and `x_(2)` satisfy the condition `x_(1)^(2)+x_(2)^(2)=5,3(x_(1)^(5)+x_(2)^(5))=11(x_(1)^(3)+x_(2)^(3))` (assume that `x_(1),x_(2)` are real)

Explanation

We have `3(x_(1)^(5)+x_(2)^(5))=11(x_(1)^(3)+x_(2)^(3))`
`implies(x_(1)^(5)+x_(2)^(5))/(x_(1)^(3)+x_(2)^(3))=11/3`
`implies((x_(1)^(2)+x_(2)^(2))(x_(1)^(3)+x_(2)^(3))-x_(1)^(2)x_(2)^(2)(x_(1)+x_(2)))/((x_(1)^(3)+x_(2)^(3)))=11/3`
`implies(x_(1)^(2)+x_(2)^(2))-(x_(1)^(2)x_(2)^(2)(x_(1)+x_(2)))/((x_(1)+x_(2))(x_(1)^(2)+x_(2)^(2)-x_(1)x_(2)))=11/3`
`[:' x_(1)^(2)+x_(2)^(2)=5]`
`implies5-(x_(1)^(2)x_(2)^(2))/(5-x_(1)^(2))=11/3`
`implies4/3=(x_(1)^(2)x_(2)^(2))/(5-x_(1)x_(2))`
`implies3x_(1)^(2)x_(2)^(2)+4x_(1)^(2)-20=0`
`implies3x_(1)^(2)x_(2)^(2)+10x_(1)x_(2)-6x_(1)x_(2)-20=0`
`implies(x_(1)x_(2)-2)(3x_(1)x_(2)+10)=0`
`:.x_(1)x_(2)=2(-10/3)`
We have `(x_(1)+x_(2))^(2)=x_(1)^(2)+x_(2)^(2)+2x_(1)x_(2)=5+2x_(1)x_(2)`
`:.(x_(1)+x_(2))^(2)=5+4=9` [ if `x_(1)x_(2)=2`]
`:.x_(1)+x_(3)=+-3`
and `(x_(1)+x_(2))^(2)=5+2(-10/3)=-5/3` [if `x_(1)x_(2)=-10/3`]
which is not possible since `x_(1),x_(2)` are real.
Thus required quadratic equations are `x^(2)+-3x+2=0`