["# Analyzing the Quadratic Equation: ( ax^2 + bx + c = 0 ) with ( a = 3 ), ( b = -12 ), ( c = 9 )", "When solving quadratic equations, selecting concrete numerical values simplifies understanding their behavior, roots, and real-world applications. Here, we explore the quadratic equation defined by ( a = 3 ), ( b = -12 ), and ( c = 9 ). This equation offers great insight into how coefficients influence the shape and solutions of parabolas.", "## Understanding the Equation", "The standard form of a quadratic equation is:", "[
\nax^2 + bx + c = 0
\n]", "With ( a = 3 ), ( b = -12 ), and ( c = 9 ), the equation becomes:", "[
\n3x^2 - 12x + 9 = 0
\n]", "Here, since ( a > 0 ), the parabola opens upwards—a key property affecting the number and nature of its roots.", "## Step 1: Simplify the Equation", "To make solving easier, we can simplify the equation by dividing all terms by ( 3 ):", "[
\nx^2 - 4x + 3 = 0
\n]", "This simplified form retains all original properties but reduces complexity.", "## Step 2: Factor the Quadratic", "We now attempt to factor ( x^2 - 4x + 3 ):", "We look for two numbers that multiply to ( 3 ) (the constant term) and add to ( -4 ) (the coefficient of ( x )).", "Those numbers are ( -3 ) and ( -1 ), since:", "[
\n-3 \ imes -1 = 3 \quad \ ext{and} \quad -3 + (-1) = -4
\n]", "Thus, the factored form is:", "[
\n(x - 3)(x - 1) = 0
\n]", "## Step 3: Solve the Equation", "Using the zero-product property:", "[
\nx - 3 = 0 \quad \ ext{or} \quad x - 1 = 0
\n]", "Solving these gives:", "[
\nx = 3 \quad \ ext{and} \quad x = 1
\n]", "So, the solutions to the equation are ( x = 1 ) and ( x = 3 ).", "## Step 4: Analyze the Roots", "- Nature of roots: Two real and distinct linear roots.
\n- Roots' location: Both roots lie on the number line at ( 1 ) and ( 3 ), located symmetrically around ( x = 2 ), the axis of symmetry for this parabola.
\n- Discriminant: Recall the discriminant ( D = b^2 - 4ac ). Plugging in values:", "[
\nD = (-12)^2 - 4(3)(9) = 144 - 108 = 36 > 0
\n]", "A positive discriminant confirms two real, distinct roots.", "## Step 5: Graph the Parabola", "- Vertex location: The axis of symmetry is at ( x = \frac{-b}{2a} = \frac{12}{6} = 2 ).
\n- Minimum value: Since the parabola opens upwards, the vertex at ( x = 2 ) is a minimum point. Evaluating:", "[
\ny = 3(2)^2 - 12(2) + 9 = 12 - 24 + 9 = -3
\n]", "- Y-intercept: Set ( x = 0 ):", "[
\ny = 3(0)^2 - 12(0) + 9 = 9
\n]", "Plotting these key features confirms the parabola crossing the x-axis at ( x = 1 ) and ( x = 3 ), with vertex at ( (2, -3) ).", "## Why This Equation Matters", "Quadratic equations like ( 3x^2 - 12x + 9 = 0 ) appear in physics, engineering, and economics—modelling motion, optimization, and cost functions. With clear integer coefficients and real roots, this example is ideal for teaching or real-world derivation without complex decimal calculations.", "---", "Summary:
\nFor ( a = 3 ), ( b = -12 ), and ( c = 9 ), the quadratic equation is ( 3x^2 - 12x + 9 = 0 ), factoring neatly to ( (x - 3)(x - 1) = 0 ), yielding solutions ( x = 1 ) and ( x = 3 ). The discriminant confirms two real roots, the parabola opens upward, and the vertex illustrates the minimum point. This simple case exemplifies how coefficients shape quadratic behavior in both theory and practice.", "---", "Keywords: quadratic equation, 3x² - 12x + 9 = 0, roots, discriminant, parabola graph, algebra, math example, solving quadratics, ( a = 3 ), ( b = -12 ), ( c = 9 )"]
