Quadratic equation

finding the discriminant of the given quadratic equation

D= (4abcd)² -4(a⁴+b⁴)(c⁴+d⁴)

D= 4{4a²b²c²d² -(a⁴c⁴+a⁴d⁴+b⁴c⁴+b⁴d⁴)

D= -4{ a⁴c⁴ +b⁴d⁴ - 2a²b²c²d² + a⁴d⁴ + b⁴c⁴ - 2 a²b²c²d² }

D= -4 { (a²c² -b²d²)² + (a²d²-b²c²)² }

thus ,

D<=0

for roots to be real
D>=0

thus for real roots
D=0 => roots are equal
Hence if roots are real they are equal

(a^4+b^4)x^2+4abcdx + (c^4+d^4) \ge 2a^2b^2 x^2 + 4abcdx + 2c^2d^2 \ \text{from AM-GM Ineq}

\ge 2 (abx + cd)^2 \ge 0

Note that

(i) Equality occurs iff a=b and c=d

(ii) There is a unique value of x for which the expression then becomes zero

q.e.d