Question

If $a,b,c,d\in R,$ then the equation $(x^2+ax-3b)(x^2-cx+b)(x^2-dx+2b)=0$ has

• A. exactly six real roots
• B. at least two real roots
• C. at least four real roots
• D. no real roots

Answer 1

The correct option is B at least two real roots

The discriminants of the given quadratic factors are
$D_1=a^2+12b,D_2=c^2-4b$ and $D_3=d^2-8b$
$D_1+D_2+D_3=a^2+c^2+d^2\ge 0$
Hence, at least one of $D_1,D_2,D_3$ is non-negative.
Therefore, the equation has at least two real roots.