Focus on the Root!

Since $(i+a)$ is a root, then $(a-i)$ is also a root. As a result, $f(x)=(x-(a+i))(x-(a-i))=x^2-2ax+(a^2+1)$ . Completing the square and substituting $f(x)$ with $y$ yields $(y-1)=1(x-a)^2$ . This means the directrix is at $y=1-0.25=0.75=2a^2-0.5a$ , as defined in the problem. Solving for a yields $a=0.75$ and $a=-0.5$ , but $a=-0.5$ is eliminated as $a>0$ as stated in the problem. As a result, the focus is at $(0.75, 1+0.25) = (0.75, 1.25)$ . Finally, $0.75+1.25=2$ which is the final answer.