Question:** In a machine learning model, the cost function is given by \( C(w) = pw^2 + qw + r \). If \( C(1) = 10 \), \( C(2) = 18 \), and \( C(3) = 30 \), find the parameters \( p \), \( q \), and \( r \).

["Understanding and Solving a Machine Learning Cost Function: A Step-by-Step Guide to Finding Parameters ( p ), ( q ), and ( r )", "In machine learning, the cost function quantifies how well a model fits the given data. Accurately determining its parameters is essential for effective training. Consider the quadratic cost function:\n[\nC(w) = pw^2 + qw + r\n]\nGiven specific values:\n[\nC(1) = 10, \quad C(2) = 18, \quad C(3) = 30\n]\nthis article walks you through solving for the unknowns ( p ), ( q ), and ( r ), a foundational exercise in model optimization.", "### Step 1: Plug in the Given Data into the Cost Function", "Start by substituting each input into the quadratic form:", "1. For ( w = 1 ):\n[\nC(1) = p(1)^2 + q(1) + r = p + q + r = 10\n]", "2. For ( w = 2 ):\n[\nC(2) = p(2)^2 + q(2) + r = 4p + 2q + r = 18\n]", "3. For ( w = 3 ):\n[\nC(3) = p(3)^2 + q(3) + r = 9p + 3q + r = 30\n]", "Now we have a system of three linear equations:\n[\n\begin{cases}\np + q + r = 10 \quad &\ ext{(1)} \\n4p + 2q + r = 18 \quad &\ ext{(2)} \\n9p + 3q + r = 30 \quad &\ ext{(3)}\n\end{cases}\n]", "### Step 2: Eliminate ( r ) to Reduce the System", "Subtract equation (1) from equation (2):\n[\n(4p + 2q + r) - (p + q + r) = 18 - 10\n\Rightarrow 3p + q = 8 \quad \ ext{(4)}\n]", "Subtract equation (2) from equation (3):\n[\n(9p + 3q + r) - (4p + 2q + r) = 30 - 18\n\Rightarrow 5p + q = 12 \quad \ ext{(5)}\n]", "### Step 3: Solve for ( p ) Using Equations (4) and (5)", "Subtract equation (4) from equation (5):\n[\n(5p + q) - (3p + q) = 12 - 8\n\Rightarrow 2p = 4 \Rightarrow p = 2\n]", "### Step 4: Solve for ( q ) Using Equation (4)", "Substitute ( p = 2 ) into equation (4):\n[\n3(2) + q = 8 \Rightarrow 6 + q = 8 \Rightarrow q = 2\n]", "### Step 5: Solve for ( r ) Using Equation (1)", "Substitute ( p = 2 ) and ( q = 2 ) into equation (1):\n[\n2 + 2 + r = 10 \Rightarrow r = 6\n]", "### Final Result: Parameters of the Cost Function", "The cost function parameters are:\n[\np = 2, \quad q = 2, \quad r = 6\n]\nThus, the cost function becomes:\n[\nC(w) = 2w^2 + 2w + 6\n]", "### Why This Matters in Machine Learning", "Understanding how to derive model parameters from cost function data is key in developing and tuning machine learning models. Calibrating ( p ), ( q ), and ( r ) ensures optimal predictions—critical for applications ranging from financial forecasting to medical diagnostics. This explicit derivation reinforces foundational math skills needed in gradient-based optimization and model evaluation.", "Tip: Tools like linear regression solvers automate this process, but knowing the manual method builds deeper intuition into how machine learning models learn and minimize error.", "---", "Keywords: machine learning cost function, quadratic cost function, derive parameters ( p ), ( q ), ( r ), solve system of equations, machine learning modeling, gradient descent prelude, ( C(w) = pw^2 + qw + r ), parameter estimation."]