The roots of the equation a4+b4x2+4abcdx+c4+d4=0

The discriminant of the given equation isD=16a2b2c2d2−4a4+b4c4+d4=−4a4+b4c4+d4−4a2b2c2d2=−4a4c4+a4d4+b4c4+b4d4−4a2b2c2d2=−4a4c4+b4d4−2a2b2c2d2+a4d4+b4c4−2a2b2c2d2=−4a2c2−b2d22+a2d2−b2c22----1 If roots of the given equation are real, D≥0⇒−4a2c2−b2d22+a2d2−b2c22≥0⇒ a2c2−b2d22+a2d2−b2c22≤0⇒ a2c2−b2d22+a2d2−b2c22=0----2['.' Sum of two positive quantities cannot be negative] From (1) and (2), we get D = 0 Hence, the roots of the given quadratic equation are not different if they are real.