Question

Does there exist a quadratic equation whose coefficients are rational but both of its roots are irrational? Justify you answer.

Answer 1

Yes.
Consider the quadratic equation
`3x^(2)-x-5=0` with rational coefficients
Here a=3, b=-1, c=-5
`D=b^(2)-4ac=(-1)^(2)-4(3)(-5)=61`
`becausex=(-b+-sqrtD)/(2a)=(1+-sqrt61)/(6)`
which are irrational.
We can take always the irrational roots which are canjugate in pairs for which coefficient will be rational as
`a+sqrtbanda-sqrtbor5+sqrt3,-5sqrt3` etc.