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You provided a formula that looks like:

\[
O = \frac{R}{j\omega L_2 + \omega^2 L_2 L_1 R + R}
\]

You requested to solve for \( \omega \) (omega). Let's rearrange the equation to express \( \omega \) explicitly.

We start by multiplying both sides by the denominator to eliminate the fraction:

\[
O \cdot \left( j\omega L_2 + \omega^2 L_2 L_1 R + R \right) = R
\]

Then expand this equation:

\[
O \left( j\omega L_2 + \omega^2 L_2 L_1 R + R \right) = R
\]

Now, we can solve for \( \omega \). Let me handle the algebra steps explicitly for you.The solutions for \( \omega \) are:

\[
\omega_1 = \frac{-i L_2 O + \sqrt{L_2 O \left(-4 L_1 O R^2 + 4 L_1 R^2 - L_2 O\right)}}{2 L_1 L_2 O R}
\]

\[
\omega_2 = \frac{-i L_2 O - \sqrt{L_2 O \left(-4 L_1 O R^2 + 4 L_1 R^2 - L_2 O\right)}}{2 L_1 L_2 O R}
\]

These two solutions represent the potential values of \( \omega \) given the equation.

Would you like further clarification or a specific step explained?

Here are some related questions:
1. How does the presence of complex terms impact the nature of \( \omega \)?
2. What are the physical interpretations of the terms \( L_1 \), \( L_2 \), and \( R \) in this equation?
3. Could you apply these results to a specific scenario, such as an RLC circuit?
4. How would you determine the resonance frequency from this equation?
5. How does the factor of \( O \) affect the solution for \( \omega \)?

**Tip**: When solving for frequencies in complex circuits, it's essential to account for both the real and imaginary parts to capture both resonance and phase behavior.

Learn how to solve for omega (ω) in a complex circuit equation involving inductances (L1, L2), resistance (R), and complex terms. Step-by-step solution suitable for university-level electrical engineering or physics students.

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