Question

If $p, q, r, s \in R$, then equation $\left(x^2+p x+3 q\right)\left(-x^2+r x+q\right)\left(-x^2+S x-2 q\right)=0$ has

• A. 6 real roots.
• B. at least two real roots.
• C. 2 real and 4 imaginary roots.
• D. 4 real and 2 imaginary roots.

Answer 1

Answer: (B)

$ D_1=p^2-12 q$
$D_2=r^2+4 q $
$D_3=s^2-8 q$
Case-I : $q <0 \Rightarrow D _1>0, D _3>0$ But $D _2$ not defined.
Case-II : $q =0 \Rightarrow D _1>0, D _2>0, D _3>0^2$
Case-III : $q >0 \Rightarrow D _2>0$ but $D _1$ and $D _3$ not defined. Hence, at least two real roots.