To create a quadratic equation for the company's daily profit based on the selling price of dresses, we can use the general form of a quadratic equation:
\[ p(d) = ad^2 + bd + c \]
Given that the maximum daily profit is $1600 when dresses are sold for $75, we have the point \((75, 1600)\). Plugging these values into the equation:
\[ 1600 = a(75)^2 + b(75) + c \]
Similarly, when dresses are sold for $50, the profit is $1225, giving us the point \((50, 1225)\):
\[ 1225 = a(50)^2 + b(50) + c \]
We now have a system of three equations:
\[ \begin{cases} 1600 = a(75)^2 + b(75) + c \\ 1225 = a(50)^2 + b(50) + c \end{cases} \]
We also know that the quadratic relation has a maximum at the vertex, which is given by the formula \((-b/(2a), f(-b/(2a)))\). In this case, the vertex is \((-b/(2a), f(-b/(2a))) = (75, 1600)\). This provides a third equation:
\[ \frac{-b}{2a} = 75 \]
Now you can solve this system of equations to find the values of \(a\), \(b\), and \(c\), allowing you to formulate the quadratic equation for the company's daily profit.