A function \(f(x) = ax^2 + bx + c\) passes through the points (1, 2), (2, 3), and (3, 6). Find \(a + b + c\).
["## Finding (a + b + c) for the Quadratic Function Passing Through Three Points", "Quadratic functions in the form ( f(x) = ax^2 + bx + c ) are fundamental in algebra, and determining their coefficients often involves solving a system of equations derived from known data points. In this article, we explore how to find the values of ( a ), ( b ), and ( c ) when the quadratic passes through three specific points: (1, 2), (2, 3), and (3, 6). Our goal is to compute ( a + b + c ).", "### Setting Up the System of Equations", "Given the quadratic ( f(x) = ax^2 + bx + c ), and the points through which it passes:", "- At ( x = 1 ), ( f(1) = 2 ):\n [\n a(1)^2 + b(1) + c = 2 \Rightarrow a + b + c = 2 \quad \ ext{(Equation 1)}\n ]", "- At ( x = 2 ), ( f(2) = 3 ):\n [\n a(2)^2 + b(2) + c = 3 \Rightarrow 4a + 2b + c = 3 \quad \ ext{(Equation 2)}\n ]", "- At ( x = 3 ), ( f(3) = 6 ):\n [\n a(3)^2 + b(3) + c = 6 \Rightarrow 9a + 3b + c = 6 \quad \ ext{(Equation 3)}\n ]", "### Solving the System for (a), (b), and (c)", "We now solve the system:\n1. ( a + b + c = 2 )\n2. ( 4a + 2b + c = 3 )\n3. ( 9a + 3b + c = 6 )", "Subtract Equation 1 from Equation 2:\n[\n(4a + 2b + c) - (a + b + c) = 3 - 2 \Rightarrow 3a + b = 1 \quad \ ext{(Equation 4)}\n]", "Subtract Equation 2 from Equation 3:\n[\n(9a + 3b + c) - (4a + 2b + c) = 6 - 3 \Rightarrow 5a + b = 3 \quad \ ext{(Equation 5)}\n]", "Now subtract Equation 4 from Equation 5:\n[\n(5a + b) - (3a + b) = 3 - 1 \Rightarrow 2a = 2 \Rightarrow a = 1\n]", "Substitute ( a = 1 ) into Equation 4:\n[\n3(1) + b = 1 \Rightarrow b = 1 - 3 = -2\n]", "Substitute ( a = 1 ), ( b = -2 ) into Equation 1:\n[\n1 - 2 + c = 2 \Rightarrow c = 2 + 2 - 1 = 3\n]", "### Verifying the Solution", "We have:\n- ( a = 1 )\n- ( b = -2 )\n- ( c = 3 )", "Check against the original points:\n- At ( x = 1 ): ( 1 - 2 + 3 = 2 ) ✅\n- At ( x = 2 ): ( 4 - 4 + 3 = 3 ) ✅\n- At ( x = 3 ): ( 9 - 6 + 3 = 6 ) ✅", "All points are satisfied.", "### Calculating ( a + b + c )", "[\na + b + c = 1 + (-2) + 3 = 2\n]", "Interestingly, since ( f(1) = 2 = a + b + c ), this confirms our result directly.", "### Conclusion", "For the quadratic function ( f(x) = ax^2 + bx + c ) passing through (1, 2), (2, 3), and (3, 6), the coefficients are ( a = 1 ), ( b = -2 ), ( c = 3 ). Therefore,", "[\na + b + c = 2\n]", "This value also corresponds to ( f(1) ), as expected. Solving such problems efficiently helps build foundational skills in algebra and real-world function modeling.", "---\nKeywords: quadratic function, pass through points, solve system of equations, ( f(x) = ax^2 + bx + c ), find ( a + b + c ), algebra tutorial, coordinate geometry."]
