7. Given $z \in \mathrm{C}$, if the equation $x^{2}-2 z x+\frac{3}{4}+\mathrm{i}=0$ (i being the imaginary unit) has real roots, then the minimum value of the modulus of the complex number $z$ $|z|$ is $\qquad$ .
$$
2 z x=x^{2}+\frac{3}{4}+\mathrm{i} \Rightarrow 2 z=x+\frac{3}{4 x}+\frac{1}{x} \mathrm{i} \Rightarrow 4|z|^{2}=\left(x+\frac{3}{4 x}\right)^{2}+\frac{1}{x^{2}}=x^{2}+\frac{25}{16 x^{2}}+\frac{3}{2} \geqslant 4 \Rightarrow|z| \geqslant 1 \text {, }
$$

When the equality holds, $x= \pm \frac{\sqrt{5}}{2}$.