proof that the equation
x^2(a^2+b^2)+2x(ac+bd)+(C^2+d^2)=0 has no real roots.
the discriminant is
4(ac+bd)^2 - 4(a^2+b^2)(c^2+d^2)
junk the 4, as it does not change things, and you have
a^2c^2 + 2abcd + b^2d^2 - a^2c^2 - a^2d^2 - b^2c^2 - b^2d^2
= 2abcd - a^2d^2 - b^2c^2
= -(ad-bc)^2
since that is always negative, there are no real roots.