i need solve this equation to get final ANSWER for z2 , and i know that L=320,A=90,T=2,Z1=23    Z2=?? GIVE ME STEPS PLS

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.

i need solve this equation to get final ANSWER for z2 , and i know that L=320,A=90,T=2,Z1=23 Z2=?? GIVE ME STEPS PLS

Follow a step-by-step solution to find Z2 in the equation L = 2A + (T/2)(Z1 + Z2) + (T^2 / 4π^2A)(Z1 - Z2)^2 with L=320, A=90, T=2, and Z1=23. This detailed guide covers advanced algebraic concepts, including quadratic equations and substitution. Suitable for students in Grades 10-12.

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