Answer
$2$ different irrational solutions that are conjugates.
Work Step by Step
$ x^{2}-5x+3=0\qquad$.... $a=1,\ b=-5,\ c=3$
$ b^{2}-4ac\qquad$....substitute $b$ for $-5,\ a$ for $1$ and $c$ for $3$
$=(-5)^{2}-4\cdot 1\cdot 3$
$=25-12$
$=13$
Since the discriminant is a positive number that is not a perfect square, there are $2$ different irrational solutions that are conjugates.