A quadratic function \( f(x) = ax^2 + bx + c \) has roots at \( x = 2 \) and \( x = -3 \). If \( f(1) = 6 \), find the value of \( a \).
["Title: How to Find the Coefficient ( a ) in a Quadratic Function Given Roots and a Point", "Meta Description: Learn how to determine the leading coefficient ( a ) of a quadratic function ( f(x) = ax^2 + bx + c ) when given its roots and a function value. Solving for ( a ) when ( f(x) ) has roots at ( x = 2 ) and ( x = -3 ), and ( f(1) = 6 ).", "---", "If you’ve ever worked with quadratic equations, you know that knowing the roots and a function value helps unlock important coefficients—especially ( a ). In this article, we’ll explore how to find the coefficient ( a ) in a quadratic function ( f(x) = ax^2 + bx + c ) when the roots are given as ( x = 2 ) and ( x = -3 ), and ( f(1) = 6 ).", "### The Power of Factored Form", "Quadratic functions with known roots can be written in factored form:\n[\nf(x) = a(x - r)(x - s)\n]\nwhere ( r ) and ( s ) are the roots. Given the roots ( x = 2 ) and ( x = -3 ), we write:\n[\nf(x) = a(x - 2)(x + 3)\n]\nHere, ( a ) is the unknown we need to find.", "### Expand the Factored Form", "To find ( a ), first expand the expression:\n[\nf(x) = a(x - 2)(x + 3) = a\left(x^2 + 3x - 2x - 6\right) = a(x^2 + x - 6)\n]\nSo,\n[\nf(x) = ax^2 + ax - 6a\n]", "This expanded form matches the standard quadratic ( f(x) = ax^2 + bx + c ), where:\n- ( b = a )\n- ( c = -6a )", "### Use the Given Value ( f(1) = 6 )", "We now use the condition that ( f(1) = 6 ). Substitute ( x = 1 ) into the expanded function:\n[\nf(1) = a(1)^2 + a(1) - 6a = a + a - 6a = -4a\n]\nSet this equal to 6:\n[\n-4a = 6\n]", "### Solve for ( a )", "Solving the equation:\n[\na = \frac{6}{-4} = -\frac{3}{2}\n]", "---", "### Final Answer", "The value of ( a ) is:\n[\n\boxed{-\frac{3}{2}}\n]", "This means the quadratic function is:\n[\nf(x) = -\frac{3}{2}(x - 2)(x + 3)\n]\nwhich confirms roots at ( x = 2 ) and ( x = -3 ), and satisfies ( f(1) = 6 ).", "Understanding how to extract coefficients from roots and function values is essential for analyzing quadratic models in science, engineering, and economics. Whether you're fitting data or solving equations, mastering this technique saves time and strengthens comprehension.", "---", "Keywords: quadratic function, roots of a quadratic, find ( a ), vertex form, plugging in values, ( f(x) = ax^2 + bx + c ), solving for coefficient, ( f(1) = 6 )", "Search Intent: Students and learners seeking step-by-step guidance to determine the leading coefficient ( a ) of a quadratic with known roots and a point."]
