Solving a polynomial equation with a condition of equality on roots

Let the following equation have two equal roots:

f[x_] := x^3 - p x^2 + q x - r 

And I want to find out what the three roots are. Not knowing how to put this condition within a Solve or Reduce, I have to rely on transcribing some algebra into MMA.So if the above equation has two equal roots, then

fr[x_] := 3 x^2 - 2 p x + q

has one of them, out of a theorem on the depression of equations.If I put this into Solve this leads to:

Solve[x^3 - p x^2 + q  x - r == 0   && 3 x^2 - 2 p  x + q == 0, x]

No error message but MMA is unable to give here one the the duplicate solution.So I put more algebra into the code:

PolynomialRemainder[f[x], fr[x], x]    (* the remainder must he equal to zero *)rootduplicate = x /. Solve[% == 0, x]  (* gives the value of the two equal roots *)

The third root is within the third factor of f[x]:

PolynomialQuotient[f[x], (x - rootduplicate[[1]])^2, x] // FullSimplifySolve[% == 0, x]root3 = x /. {%}

Let's check it up with an example:

Solve[(x^3 - p x^2 + q  x - r /. {p -> 4, q -> 5, r -> 2}) ==  0   && (3 x^2 - 2 p  x + q /. {p -> 4, q -> 5}) == 0, x]    (* now MMA finds a double root *)Solve[(x^3 - p x^2 + q  x - r /. {p -> 4, q -> 5, r -> 2}) ==  0   , x] rootduplicate /. {p -> 4, q -> 5, r -> 2}root3 /. {p -> 4, q -> 5, r -> 2}root3 == {(p - 2 rootduplicate)} // FullSimplify (* another verification out of algebra   - is there a way with MMA to find out this relation without hardcoding it? *)

My questions:

• Why MMA (V8) is unable to find a solution in my first Solve ?
• Is there a better way relying less on algebra to get these solutions?