A quadratic equation \( ax^2 + bx + c = 0 \) has roots 3 and -2. Find the value of \( a + b + c \) if \( a = 1 \).

["Optimizing Your Understanding: Solving Quadratic Equations with Known Roots", "Finding the roots of a quadratic equation can simplify complex problems in mathematics, physics, and engineering. When given that the quadratic equation ( ax^2 + bx + c = 0 ) has roots ( 3 ) and ( -2 ), and knowing ( a = 1 ), let’s explore how to determine the coefficients and compute the critical sum ( a + b + c ).", "### Understanding Quadratic Roots and Their Impact on Equation Form", "A quadratic equation with roots ( r_1 ) and ( r_2 ) can be expressed in factored form as:\n[\na(x - r_1)(x - r_2) = 0\n]\nSubstituting the given roots ( 3 ) and ( -2 ), and ( a = 1 ):\n[\n(x - 3)(x + 2) = 0\n]", "### Expanding the Factored Form", "Multiply the binomials:\n[\n(x - 3)(x + 2) = x^2 + 2x - 3x - 6 = x^2 - x - 6\n]", "So, the quadratic equation becomes:\n[\nx^2 - x - 6 = 0\n]", "From this, we identify the coefficients:\n[\na = 1, \quad b = -1, \quad c = -6\n]", "### Calculating ( a + b + c )", "Now, compute the sum:\n[\na + b + c = 1 + (-1) + (-6) = 1 - 1 - 6 = -6\n]", "Final result:\n[\na + b + c = -6\n]", "### Why This Matters", "Understanding how roots determine coefficients helps solve real-world problems quickly—like projectile motion, optimization, or circuit design—without solving from scratch. With ( a = 1 ), expressing the equation via roots ensures accuracy and efficiency, reducing computational steps.", "Key takeaway: For quadratic equations ( ax^2 + bx + c = 0 ), given roots ( r_1, r_2 ) and ( a = 1 ), the sum ( a + b + c ) directly reflects the constant term after expansion. This method streamlines analysis and strengthens conceptual grasp of algebra.", "Keywords: quadratic equation, roots of quadratic, find ( a + b + c ), factor quadratic, solve quadratic, ( ax^2 + bx + c = 0 ), equation coefficients, algebraic identities."]