Solving $KL^{2C} - LC^{2C} = (B-C)C^{2C} - C^{2C}(T+KG^2) - KP^2$ for $L$

Question

How can I solve this equation for $L$. Everything else is known number: $$KL^{2C} - LC^{2C} = (B-C)C^{2C} - C^{2C}(T+KG^2) - KP^2$$

Answer 1

If $K = 0$ this is a line equation in $L$, so let's assume $K \ne 0$. Let's also assume $C$ is a non-negative integer.

Then you are looking at $$L^{2C} - aL -b = 0$$ and for $C>2$ you will not necessarily have analytic solutions.

For $C=0$, it is also linear in $L$. For $C=1$, the quadratic formula applies, and for $C=2$, you have to solve a 4-degree equation, for which you can find closed analytic formulae akin to the quadratic formula...