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for roots to be non real the value the discriminant must be less than 0
now
by using. b²-4ac<0
(2(AC+bd))² - 4(a²+b²)(c²+d²) < 0
4a²c²+4b²d²+8abcd-4(a²c²+a²d²+b²c²+b²d²)<0
8abcd<4a²d²+4b²c²
2abcd<a²d²+b²c²
also ad is not equal to bc thus the roots are purely imaginary
now
by using. b²-4ac<0
(2(AC+bd))² - 4(a²+b²)(c²+d²) < 0
4a²c²+4b²d²+8abcd-4(a²c²+a²d²+b²c²+b²d²)<0
8abcd<4a²d²+4b²c²
2abcd<a²d²+b²c²
also ad is not equal to bc thus the roots are purely imaginary
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