Repeated roots of biquadratic equation What is condition for repeated roots of the fourth order polynomial 
$$ x^4 + a x^3 + b x^2 + c x + d = 0 ?$$

A polynomial has roots with multiplicity greater than one iff its discriminant is zero. 
The discriminant of your generic fourth-degree polynomial is:
$$a^2 b^2 c^2-4 b^3 c^2-4 a^3 c^3+18 a b c^3-27 c^4-4 a^2 b^3 d+16 b^4 d+18 a^3 b c d-80 a b^2 c d-6 a^2 c^2 d+144 b c^2 d-27 a^4 d^2+144 a^2 b d^2-128 b^2 d^2-192 a c d^2+256 d^3.$$
