1. **State the problem:** We are given the profit function $$y = -x^2 + 84x - 630$$ where $y$ is the profit and $x$ is the selling price per widget. We need to find the maximum profit the company can make. 2. **Identify the type of function:** This is a quadratic function in the form $$y = ax^2 + bx + c$$ with $$a = -1$$, $$b = 84$$, and $$c = -630$$. Since $$a < 0$$, the parabola opens downward, so the vertex represents the maximum point. 3. **Formula for the vertex:** The $x$-coordinate of the vertex is given by $$x = -\frac{b}{2a}$$. 4. **Calculate the $x$-coordinate of the vertex:** $$x = -\frac{84}{2 \times (-1)} = -\frac{84}{-2} = 42$$ 5. **Calculate the maximum profit by substituting $x=42$ into the profit function:** $$y = -(42)^2 + 84 \times 42 - 630$$ $$y = -1764 + 3528 - 630$$ $$y = 1134$$ 6. **Interpretation:** The maximum profit the company can make is 1134 (to the nearest dollar).