According to the fundamental theorem of algebra, a polynomial has the number of zeros equivalent to the degree of the polynomial.
A quadratic equation is of degree 2 and hence has 2 zeros.
A complex zero is of the form
![a\pm bi](https://tex.z-dn.net/?f=a%5Cpm%20bi)
which represents two zeros: a + bi and a - bi.
Thus, since a quadratic equation has only 2 zeros, a quadratic equation with a complex zero already have two zeros and hence will not have any other zero.