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 \[b=0\]and\[d=0\] 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\[\frac{c}{a},\frac{-c}{a}\]must be positive, so two roots must be real.You need to login to perform this action.
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