## Problem Description and Formulation

The problem is an optimization problem with two variables: 'milligrams of iron' and 'milligrams of potassium'. The objective function to be minimized is:

\[ 5 \times (\text{milligrams of iron})^2 + 4 \times (\text{milligrams of iron}) \times (\text{milligrams of potassium}) + 3 \times (\text{milligrams of potassium})^2 + 8 \times (\text{milligrams of iron}) + 7 \times (\text{milligrams of potassium}) \]

The problem has several constraints:

1. The immune support index for milligrams of iron is 2.
2. The muscle growth index for milligrams of iron is 10.
3. Milligrams of potassium have an immune support index of 9.
4. Milligrams of potassium each have a muscle growth index of 3.
5. The total combined immune support index from milligrams of iron squared plus milligrams of potassium squared should be greater than or equal to 48.
6. The total combined immune support index from milligrams of iron plus milligrams of potassium should be no less than 48.
7. The total combined muscle growth index from milligrams of iron plus milligrams of potassium must be at least 18.
8. The total combined muscle growth index from milligrams of iron and milligrams of potassium must be 18 at minimum.
9. \(-6 \times (\text{milligrams of iron}) + 5 \times (\text{milligrams of potassium}) \geq 0\).
10. The total combined immune support index from milligrams of iron plus milligrams of potassium should be at maximum 56.
11. The total combined muscle growth index from milligrams of iron plus milligrams of potassium should be less than or equal to 55.
12. There can be a non-integer amount of milligrams of iron.
13. There must be an integer number of milligrams of potassium.

## Gurobi Code Formulation

```python
import gurobi

def optimize_problem():
    # Create a new Gurobi model
    model = gurobi.Model()

    # Define variables
    iron = model.addVar(lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY, name="iron", vtype=gurobi.GRB.CONTINUOUS)
    potassium = model.addVar(lb=0, name="potassium", vtype=gurobi.GRB.INTEGER)

    # Objective function
    model.setObjective(5 * iron**2 + 4 * iron * potassium + 3 * potassium**2 + 8 * iron + 7 * potassium, gurobi.GRB.MINIMIZE)

    # Constraints
    model.addConstr(2 * iron <= 109, name="immune_support_index_iron")
    model.addConstr(10 * iron <= 65, name="muscle_growth_index_iron")
    model.addConstr(9 * potassium <= 109, name="immune_support_index_potassium")
    model.addConstr(3 * potassium <= 65, name="muscle_growth_index_potassium")

    model.addConstr(iron**2 + potassium**2 >= 48, name="immune_support_index_combined_squared")
    model.addConstr(2 * iron + 9 * potassium >= 48, name="immune_support_index_combined")
    model.addConstr(10 * iron + 3 * potassium >= 18, name="muscle_growth_index_combined")
    model.addConstr(-6 * iron + 5 * potassium >= 0, name="linear_constraint")
    model.addConstr(2 * iron + 9 * potassium <= 56, name="immune_support_index_combined_upper")
    model.addConstr(10 * iron + 3 * potassium <= 55, name="muscle_growth_index_combined_upper")

    # Solve the problem
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal solution found.")
        print(f"Iron: {iron.varValue}")
        print(f"Potassium: {potassium.varValue}")
        print(f"Objective: {model.objVal}")
    elif model.status == gurobi.GRB.INFEASIBLE:
        print("The problem is infeasible.")
    else:
        print("The problem has no optimal solution.")

optimize_problem()
```