1. **State the problem:** We need to find the number of network devices $n$ that minimizes the total cost $C$ given by the equation $$C = 8n^2 - 176n + 1800.$$ 2. **Formula and concept:** This is a quadratic function in the form $$C = an^2 + bn + c$$ where $a=8$, $b=-176$, and $c=1800$. Since $a > 0$, the parabola opens upwards, so the vertex represents the minimum point. 3. **Find the vertex:** The $n$-value of the vertex is given by $$n = -\frac{b}{2a}.$$ 4. **Calculate:** Substitute $a=8$ and $b=-176$ into the formula: $$n = -\frac{-176}{2 \times 8} = \frac{176}{16} = 11.$$ 5. **Interpretation:** The company should install **11 network devices** to minimize the cost of establishing the network. 6. **Verify by plugging back:** Calculate cost at $n=11$: $$C = 8(11)^2 - 176(11) + 1800 = 8 \times 121 - 1936 + 1800 = 968 - 1936 + 1800 = 832.$$ This confirms the minimum cost occurs at $n=11$.