Answer
There are $2$ different irrational solutions that are conjugates.
Work Step by Step
$ x^{2}+5x=9\qquad$....add $-9$ to each side to write in form of $ax^{2}+bx+c=0$.
$ x^{2}+5x-9=0\qquad$.... $a=1,\ b=5,\ c=-9$
$ b^{2}-4ac\qquad$....substitute $b$ for $5,\ a$ for $1$ and $c$ for $-9$
$=(5)^{2}-4\cdot 1\cdot(-9)$
$=25+36$
$=61$
Since the discriminant is a positive number that is not a perfect square, there are $2$ different irrational solutions that are conjugates.