Solution: Given $ c(n) = n^2 - 3n + 2m $ and $ c(4) = 14 $, substitute $ n = 4 $:

Optimizing the Function $ c(n) = n^2 - 3n + 2m $: Solve for $ m $ Using $ c(4) = 14 $

In mathematical modeling and optimization, identifying unknown constants from known values is a common challenge. One such problem arises when given a quadratic function of the form:

$$
c(n) = n^2 - 3n + 2m
$$
and provided a specific output value at a certain input: $ c(4) = 14 $. This scenario calls for substituting the known value into the function to solve for the unknown parameter $ m $.

Understanding the Context

Step-by-Step Substitution

Start with the given function:
$$
c(n) = n^2 - 3n + 2m
$$

Substitute $ n = 4 $ and $ c(4) = 14 $:
$$
14 = (4)^2 - 3(4) + 2m
$$

Simplify the right-hand side:
$$
14 = 16 - 12 + 2m
$$
$$
14 = 4 + 2m
$$

Solution: Given $ c(n) = n^2 - 3n + 2m $ and $ c(4) = 14 $, substitute $ n = 4 $:

Key Insights

Now, isolate $ 2m $:
$$
2m = 14 - 4 = 10
$$

Divide both sides by 2:
$$
m = 5
$$

Verifying the Solution

To confirm correctness, substitute $ m = 5 $ back into the original function and evaluate at $ n = 4 $:
$$
c(4) = 4^2 - 3(4) + 2(5) = 16 - 12 + 10 = 14
$$
The result matches the given value, validating our solution.

Practical Implications

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Final Thoughts

Understanding how to substitute known values into a functional equation helps in parameter estimation, especially when modeling real-world phenomena such as cost, growth, or efficiency metrics. In this case, knowing $ c(4) = 14 $ allowed us to determine the exact value of $ m $, enabling precise predictions for the model’s behavior under similar inputs.

Conclusion

Given $ c(n) = n^2 - 3n + 2m $ and $ c(4) = 14 $, substituting $ n = 4 $ yields:
$$
m = 5
$$
This solution illustrates a fundamental technique in solving for unknown constants in algebraic expressions — a vital skill in both academic problem-solving and applied data modeling.

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Keywords: Function substitution, solve for m, $ c(n) = n^2 - 3n + 2m $, $ c(4) = 14 $, algebraic solution, mathematical modeling.