Let's solve for $Z_2$ step by step given the equation:

$L = 2A + \frac{T}{2}(Z_1 + Z_2) + \frac{T^2}{4\pi^2 A}(Z_1 - Z_2)^2$

Given values:

• $L = 320$
• $A = 90$
• $T = 2$
• $Z_1 = 23$

We need to find $Z_2$.

Step 1: Substitute the known values into the equation

Substitute the given values for $L$, $A$, $T$, and $Z_1$:

$320 = 2(90) + \frac{2}{2}(23 + Z_2) + \frac{2^2}{4\pi^2(90)}(23 - Z_2)^2$

This simplifies to:

$320 = 180 + (23 + Z_2) + \frac{4}{4\pi^2(90)}(23 - Z_2)^2$

Step 2: Simplify the equation

First, simplify the constants:

$320 = 180 + 23 + Z_2 + \frac{(23 - Z_2)^2}{\pi^2 \cdot 90}$

Simplify the left side:

$320 = 203 + Z_2 + \frac{(23 - Z_2)^2}{\pi^2 \cdot 90}$

Step 3: Isolate the quadratic term

Move the constants to the left side:

$320 - 203 = Z_2 + \frac{(23 - Z_2)^2}{\pi^2 \cdot 90}$

Simplify:

$117 = Z_2 + \frac{(23 - Z_2)^2}{\pi^2 \cdot 90}$

Step 4: Eliminate the fraction

Multiply through by $\pi^2 \cdot 90$ to eliminate the fraction:

$117 (\pi^2 \cdot 90) = Z_2 (\pi^2 \cdot 90) + (23 - Z_2)^2$

Simplify:

$10530 \pi^2 = Z_2 \pi^2 \cdot 90 + (23 - Z_2)^2$

Step 5: Expand and solve for $Z_2$

Now, you have a quadratic equation. I will expand and solve it for $Z_2$ numerically.

Let's compute it.