A) Four real roots
B) Two real roots
C) Four imaginary roots
D) None of these
Correct Answer: B
Solution :
Let all four roots are imaginary. Then roots of both equations \[P(x)=0\]and \[Q(x)=0\]are imaginary. Thus\[{{b}^{2}}-4ac<0;{{d}^{2}}+4ac<0\], So\[{{b}^{2}}+{{d}^{2}}<0\], which is impossible unless\[b=0,d=0\]. So, if \[b\ne 0\]or \[d\ne 0\] at least two roots must be real. If \[b=0,\] \[d=0\], we have the equations. \[P(x)=a{{x}^{2}}+c=0\]and\[Q(x)=-a{{x}^{2}}+c=0\] or \[{{x}^{2}}=-\frac{c}{a};{{x}^{2}}=\frac{c}{a}\] as one of \[\frac{c}{a}\] and \[-\frac{c}{a}\] must be positive, so two roots must be real.
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