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Thus, the value of \( k \) that ensures the minimum cost is zero is \(\boxed{12}\).
An AI programmer is optimizing a model's performance using a quadratic cost function \( f(x) = ax^2 + bx + c \). If the vertex of this parabola corresponds to the minimum point at \( x = 3 \) and the cost at the vertex is \(-5\), find the relationship between \( a \) and \( c \) given \( b = -6 \).
The vertex form of a quadratic function is \( f(x) = a(x - h)^2 + k \), where \( (h, k) \) is the vertex. Given \( h = 3 \) and \( k = -5 \), we have:

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