# If $2-i$ is a root of the eqn., $ax^2+12x+b$, where $a$ and $b$ are real then the value of $ab$ = ?

(A) 45 (B) 25 (C) 15 (D) -45

Toolbox:
• complex roots of an equation occur in conjugate pairs.
If $2-i$ is a root of $ax^2+12x+b$, then $2+i$ is the other root.
Sum of the roots $=\large-\frac{12}{a}$$=(2+i)+(2-i)=4 \Rightarrow\:a=-3 Product of the roots =\large\frac{b}{a}$$=(2+i)(2-i)=5$
$\Rightarrow\:b=5a=-15$
$\Rightarrow\:ab=45$
edited May 22, 2014