The 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: Finding Coefficient ( a ) of a Quadratic Function Given Roots and a Function Value", "When working with quadratic functions, understanding how the roots determine the equation and how a known function value helps find coefficients is essential. In this article, we explore how to determine the coefficient ( a ) in the quadratic function\n[ f(x) = ax^2 + bx + c ]\nwhen the function has roots at ( x = 2 ) and ( x = -3 ), and satisfies ( f(1) = 6 ).", "---", "### Step 1: Express the Quadratic Using Its Roots", "Since ( f(x) ) has roots at ( x = 2 ) and ( x = -3 ), it can be written in factored form:\n[ f(x) = a(x - 2)(x + 3) ]\nHere, ( a ) is the leading coefficient that we need to determine.", "---", "### Step 2: Expand the Factored Form (Optional)", "Expanding the expression helps convert it to standard form:\n[\nf(x) = a(x - 2)(x + 3) = a\left(x^2 + 3x - 2x - 6\right) = a(x^2 + x - 6)\n]\nSo,\n[ f(x) = ax^2 + ax - 6a ]\nFrom this, we identify:\n- ( b = a )\n- ( c = -6a )", "---", "### Step 3: Use the Given Value ( f(1) = 6 )", "Substitute ( x = 1 ) into the expanded form:\n[\nf(1) = a(1)^2 + a(1) - 6a = a + a - 6a = -4a\n]\nBut we know ( f(1) = 6 ), so:\n[\n-4a = 6\n]", "---", "### Step 4: Solve for ( a )", "Divide both sides by (-4):\n[\na = -\frac{6}{4} = -\frac{3}{2}\n]", "---", "### Conclusion", "The value of the coefficient ( a ) in the quadratic function ( f(x) = ax^2 + bx + c ), given that it has roots at ( x = 2 ) and ( x = -3 ) and satisfies ( f(1) = 6 ), is:\n[\n\boxed{-\frac{3}{2}}\n]", "This example illustrates the powerful connection between roots, function values, and coefficients—key tools for solving quadratic equations efficiently.\nKeywords: quadratic function, roots, coefficients, function value, ( f(x) = ax^2 + bx + c ), solve for ( a ), algebra basics."]