Let two numbers have arithmetic mean 9 and  geometric mean 4. Then, these numbers are the roots of the quadratic equation 

Let $a$ and $\beta$ be two numbers whose arithmetic mean is 9 and geometric mean is 4. 

Then. $\alpha +\beta =18\text{ and }\alpha \beta =16$

$\therefore$ Required equation is 

$$\begin{gathered} x^2-(\alpha +\beta )x+\left(\alpha \beta \right)=0 \\ \Rightarrow x^2-18x+16=0 \end{gathered}$$