A quadratic equation \( ax^2 + bx + c = 0 \) has roots 3 and -2. If \( a = 1 \), find \( b \) and \( c \), and determine the value of the expression \( b^2 - 4ac \).
["Understanding Quadratic Roots: Finding ( b ), ( c ), and the Discriminant", "When solving a quadratic equation of the form ( ax^2 + bx + c = 0 ), a powerful technique involves using the roots of the equation. In this article, we explore a specific case where the roots are ( 3 ) and ( -2 ), and the leading coefficient ( a = 1 ). We’ll determine the coefficients ( b ) and ( c ), and calculate the discriminant ( b^2 - 4ac )—a key expression that reveals insights about the nature of the roots.", "### Given Data:\n- Roots: ( x = 3 ) and ( x = -2 )\n- Leading coefficient: ( a = 1 )", "---", "### Step 1: Use the Root Information to Write the Equation", "If ( x = 3 ) and ( x = -2 ) are roots of ( ax^2 + bx + c = 0 ), then the quadratic can be expressed in factored form:", "[\na(x - 3)(x + 2) = 0\n]", "Since ( a = 1 ), the equation simplifies to:", "[\n(x - 3)(x + 2) = 0\n]", "Now expand the product:", "[\nx^2 + 2x - 3x - 6 = x^2 - x - 6\n]", "So the quadratic equation is:", "[\nx^2 - x - 6 = 0\n]", "---", "### Step 2: Identify coefficients ( a ), ( b ), and ( c )", "By comparing ( x^2 - x - 6 = 0 ) with ( ax^2 + bx + c = 0 ), we find:", "- ( a = 1 )\n- ( b = -1 )\n- ( c = -6 )", "---", "### Step 3: Compute the Discriminant ( b^2 - 4ac )", "The discriminant is a crucial value that tells us about the nature of the roots:", "- If ( b^2 - 4ac > 0 ): two distinct real roots\n- If ( b^2 - 4ac = 0 ): one real root (repeated)\n- If ( b^2 - 4ac < 0 ): no real roots", "Plug in the values:", "[\nb^2 - 4ac = (-1)^2 - 4(1)(-6) = 1 + 24 = 25\n]", "Since ( 25 > 0 ), the equation has two distinct real roots, which matches the given roots ( 3 ) and ( -2 ).", "---", "### Summary", "- With ( a = 1 ), and roots 3 and -2, the quadratic equation becomes:\n [\n x^2 - x - 6 = 0\n ]\n- Therefore, ( b = -1 ), ( c = -6 )\n- The discriminant ( b^2 - 4ac = 25 ), confirming real and distinct roots", "Understanding the relationship between roots, coefficients, and the discriminant empowers students and learners to solve quadratic equations efficiently and interpret solution behavior—key skills in algebra and beyond."]
