Step-by-Step Solution
Key Concept: Use Vieta's formulas to relate sums and products of roots to coefficients, then compute the required expression.
Replace $z$ by $t + \frac{1+z}{z}$ in the equation $7z^2 + 1 = 0$. This gives $4\left(\frac{1+z}{z}\right)^2 - 2\left(\frac{1+z}{z}\right) + 4 = 0$, which simplifies to $7z^2 + 1 = 0$. Computing $\frac{\alpha^2 + \beta^2}{2\alpha^2\beta^2}$ where $\alpha$ and $\beta$ are roots: $\frac{\alpha^2 + \beta^2}{2\alpha^2\beta^2} = \frac{1}{2} \cdot \frac{(\alpha+\beta)^2 - 2\alpha\beta}{(\alpha\beta)^2} = \frac{1}{2} + 1 = 1.5$.
Correct Answer: 1.5