Solution:** We are given the quadratic function \( P(x) = ax^2 + bx + c \) and the points \( P(1) = 6 \), \( P(2) = 11 \), and \( P(3) = 18 \). We can set up the following system of equations by substituting the known values:
["Solving Quadratic Functions Using Known Points: How to Use Given Values to Find ( a ), ( b ), and ( c )", "When given a quadratic function of the form ( P(x) = ax^2 + bx + c ), and specific values like ( P(1) = 6 ), ( P(2) = 11 ), and ( P(3) = 18 ), you can determine the coefficients ( a ), ( b ), and ( c ) by setting up a system of equations. This approach not only reinforces algebraic understanding but also makes solving real-world problems with quadratics much easier. In this article, we’ll walk through how to derive and solve this system step by step.", "---", "### The Quadratic Model and Substitution", "Start with the general quadratic form:\n[\nP(x) = ax^2 + bx + c\n]", "Using the given points, substitute ( x = 1 ), ( x = 2 ), and ( x = 3 ) to form equations:", "- For ( x = 1 ):\n[\na(1)^2 + b(1) + c = 6 \quad \Rightarrow \quad a + b + c = 6 \quad \ ext{(Equation 1)}\n]", "- For ( x = 2 ):\n[\na(2)^2 + b(2) + c = 11 \quad \Rightarrow \quad 4a + 2b + c = 11 \quad \ ext{(Equation 2)}\n]", "- For ( x = 3 ):\n[\na(3)^2 + b(3) + c = 18 \quad \Rightarrow \quad 9a + 3b + c = 18 \quad \ ext{(Equation 3)}\n]", "---", "### Setting Up the System of Equations", "You now have the system:\n[\n\begin{cases}\na + b + c = 6 \\n4a + 2b + c = 11 \\n9a + 3b + c = 18\n\end{cases}\n]", "This system allows you to eliminate variables step by step.", "---", "### Step 1: Eliminate ( c )", "Subtract Equation 1 from Equation 2:\n[\n(4a + 2b + c) - (a + b + c) = 11 - 6 \quad \Rightarrow \quad 3a + b = 5 \quad \ ext{(Equation 4)}\n]", "Subtract Equation 2 from Equation 3:\n[\n(9a + 3b + c) - (4a + 2b + c) = 18 - 11 \quad \Rightarrow \quad 5a + b = 7 \quad \ ext{(Equation 5)}\n]", "---", "### Step 2: Solve for ( a ) and ( b )", "Now subtract Equation 4 from Equation 5:\n[\n(5a + b) - (3a + b) = 7 - 5 \quad \Rightarrow \quad 2a = 2 \quad \Rightarrow \quad a = 1\n]", "Substitute ( a = 1 ) into Equation 4:\n[\n3(1) + b = 5 \quad \Rightarrow \quad b = 2\n]", "---", "### Step 3: Solve for ( c )", "Substitute ( a = 1 ), ( b = 2 ) into Equation 1:\n[\n1 + 2 + c = 6 \quad \Rightarrow \quad c = 3\n]", "---", "### Final Quadratic Function", "The quadratic function satisfying the given points is:\n[\nP(x) = x^2 + 2x + 3\n]", "---", "### Why This Method Works", "By substituting known points into the quadratic form, we transform abstract function behavior into concrete equations. Solving the resulting linear system reveals the precise coefficients that define the curve. This method applies universally to any quadratic with given functional values at distinct ( x )-points.", "---", "Conclusion", "Understanding how to derive and solve systems of equations from function values is essential in mathematics, physics, economics, and engineering. This framework enables you to find the exact quadratic expression—like ( P(x) = x^2 + 2x + 3 )—and equips you to tackle similar problems involving real-world modeling, optimization, and curve fitting.", "---", "SEO Keywords: quadratic function, solve quadratic, system of equations, find ( a ), ( b ), ( c ), quadratic modeling, ( P(x) = ax^2 + bx + c ), algebraic substitution, coordinate geometry, function determination."]
