4Question: A chemist models the efficiency of a new green catalyst with the function $ f(x) = x^2 - 6x + k $, where $ x $ represents reaction temperature and $ k $ is a parameter related to material properties. If the catalyst achieves peak efficiency when $ x = 3 $, what is the value of $ k $ such that $ f(3) = 0 $? - High Altitude Science
To determine the value of $ k $ that ensures peak efficiency at $ x = 3 $, we begin by analyzing the given function:
Mar 11, 2026
To determine the value of $ k $ that ensures peak efficiency at $ x = 3 $, we begin by analyzing the given function:
$$
f(x) = x^2 - 6x + k
$$
Peak efficiency occurs at $ x = 3 $, and we are told that $ f(3) = 0 $. This implies that $ x = 3 $ is a root of the function. Substituting $ x = 3 $ into the equation:
Understanding the Context
$$
f(3) = (3)^2 - 6(3) + k = 0
$$
Simplify the expression:
$$
9 - 18 + k = 0
$$
$$
-9 + k = 0
$$
Key Insights
$$
k = 9
$$
Thus, the value of $ k $ that ensures peak efficiency at $ x = 3 $ is:
$$
\boxed{9}
$$
