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=0Hence, the roots of the given quadratic equation are notdifferent if they are real.