If the zeroes of the quadratic polynomial $a{x}^2+bx+c,a\ne 0$, are equal, then:

Given, the quadratic polynomial is

$a{x}^2+bx+c,a\ne 0$ is not equal to zero. The zeroes of the quadratic polynomial are equal. 

Since the zeroes of the polynomial are equal, the value of the discriminant is 0.

$$\begin{gathered} i.e b^2-4ac=0 \\ \Rightarrow b^2= 4ac \\ {b}^2 cannot be negative as the square of any number is always positive and R.H.S is also positive. \\ Hence, a and c will have same signs \end{gathered}$$