Question

If $2+i$ and $\sqrt{5}-2i$ are the roots of the equation $(x^2+ax+b)(x^2+cx+d)=0,$ where $a,b,c,d$ are real constants, then product of all roots of the equation is

• A. $40$
• B. $9\sqrt{5}$
• C. $45$
• D. $35$

Answer 1

The correct option is C $45$

Since, imaginary roots always appear in conjugate pairs when the coefficients are real.
$2-i$ and $\sqrt{5}+2i$ must be other roots of the equation.

Product of all roots
$=(2+i)(2-i)(\sqrt{5}+2i)(\sqrt{5}-2i)$
$=(4-i^2)(5-4i^2)$
$=5\times 9$
$=45$