Let $$z$$ be a complex number with non-zero imaginary part. If

$$\frac{2 + 3z + 4z^2}{2 - 3z + 4z^2}$$

is a real number, then the value of $$|z|^2$$ is ______.

Let the given quotient be a real number and denote it by a real constant $$k$$:

$$\frac{\,2 + 3z + 4z^{2}\,}{\,2 - 3z + 4z^{2}\,}=k,\qquad k\in\mathbb{R}$$

Cross-multiplying gives a quadratic equation in $$z$$:

$$2 + 3z + 4z^{2}=k\,(2 - 3z + 4z^{2})$$

Bring all terms to the left:

$$4z^{2}(1-k)+3z(1+k)+2(1-k)=0 \qquad -(1)$$

The coefficients $$4(1-k),\;3(1+k),\;2(1-k)$$ are real because $$k$$ is real. Equation $$(1)$$ is therefore a quadratic with real coefficients.

Since $$z$$ has a non-zero imaginary part, its complex conjugate $$\bar z$$ is also a root of the same quadratic. Thus the two roots of $$(1)$$ are $$z$$ and $$\bar z$$.

For a quadratic $$az^{2}+bz+c=0$$, the product of the roots equals $$\dfrac{c}{a}$$ (Vieta’s formula). Comparing $$(1)$$ with $$az^{2}+bz+c=0$$ we have

$$a=4(1-k),\quad b=3(1+k),\quad c=2(1-k)$$

Hence

$$z\bar z=\frac{c}{a}=\frac{2(1-k)}{4(1-k)}=\frac12$$

But $$z\bar z=|z|^{2}$$, so

$$|z|^{2}=\frac12$$

Therefore the required value is 0.50.