9. Given $z \in \mathbf{C}$, if the equation $4 x^{2}-8 z x+4 \mathrm{i}+3=0$ (where $\mathrm{i}$ is the imaginary unit) has real roots, then the minimum value of the modulus of the complex number $z$ $|z|$ is $\qquad$ .
Let $z=a+b \mathrm{i}$, then $4 x^{2}-8 x(a+b \mathrm{i})+4 \mathrm{i}+3=0 \Rightarrow\left\{\begin{array}{l}4 x^{2}-8 a x+3=0, \\ -8 b x+4=0\end{array}\right.$ $\Rightarrow\left\{\begin{array}{l}a=\frac{4 x^{2}+3}{8 x}, \\ b=\frac{4}{8 x}\end{array} \Rightarrow a^{2}+b^{2}=\frac{1}{64 x^{2}}\left[\left(4 x^{2}+3\right)^{2}+16\right]=\frac{1}{64 x^{2}}\left(16 x^{4}+24 x^{2}+25\right)\right.$ $=\frac{1}{64}\left(16 x^{2}+\frac{25}{x^{2}}+24\right) \geqslant \frac{1}{64}(2 \sqrt{16 \cdot 25}+24)=1$, equality holds when $x^{2}=\frac{5}{4}$. Therefore, the minimum value of the modulus of the complex number $z$ is 1.