Here, \( a = 3 \), \( b = -12 \), and \( c = 9 \).

["Understanding Quadratic Equations: A Closer Look at the Case Where ( a = 3 ), ( b = -12 ), and ( c = 9 )", "When solving quadratic equations, identifying the coefficients ( a ), ( b ), and ( c ) is essential for applying the quadratic formula and interpreting the equation’s behavior. In this article, we analyze a specific quadratic equation with the values ( a = 3 ), ( b = -12 ), and ( c = 9 ). This example highlights key features such as the discriminant, roots, and the graph’s shape, making it a valuable teaching example in algebra.", "### The Quadratic Equation in Standard Form", "The standard quadratic equation is written as:\n[\nax^2 + bx + c = 0\n]", "For our example:\n[\n3x^2 - 12x + 9 = 0\n]\nHere, ( a = 3 ), ( b = -12 ), and ( c = 9 ), all given constants important for analysis.", "---", "### Step 1: Compute the Discriminant", "The discriminant, ( D = b^2 - 4ac ), determines the nature of the roots:", "[\nD = (-12)^2 - 4(3)(9) = 144 - 108 = 36\n]", "Since ( D = 36 > 0 ), the quadratic equation has two distinct real roots. This means the parabola crosses the x-axis at two points.", "---", "### Step 2: Find the Roots Using the Quadratic Formula", "The roots are found using:\n[\nx = \frac{-b \pm \sqrt{D}}{2a}\n]", "Substituting values:\n[\nx = \frac{-(-12) \pm \sqrt{36}}{2(3)} = \frac{12 \pm 6}{6}\n]", "Calculating both solutions:", "- ( x_1 = \frac{12 + 6}{6} = \frac{18}{6} = 3 )\n- ( x_2 = \frac{12 - 6}{6} = \frac{6}{6} = 1 )", "So, the solutions are ( x = 3 ) and ( x = 1 ). These points are where the quadratic graph intersects the x-axis.", "---", "### Step 3: Determine the Parabola’s Direction and Vertex", "Since ( a = 3 > 0 ), the parabola opens upward, confirming two distinct real roots and a U-shaped curve.", "To find the vertex, use the formula for the x-coordinate of the vertex:\n[\nx_v = -\frac{b}{2a} = -\frac{-12}{2(3)} = \frac{12}{6} = 2\n]", "Substitute back to find ( y_v ):\n[\ny_v = 3(2)^2 - 12(2) + 9 = 12 - 24 + 9 = -3\n]", "Thus, the vertex is at ( (2, -3) ), the lowest point on the parabola.", "---", "### Step 4: Factorizing the Quadratic", "With roots ( x = 1 ) and ( x = 3 ), the equation can also be expressed in factored form:\n[\na(x - r_1)(x - r_2) = 3(x - 1)(x - 3)\n]", "Expanding confirms the original equation:\n[\n3(x^2 - 4x + 3) = 3x^2 - 12x + 9\n]", "---", "### Why This Example Matters", "- Discriminant insight: The positive discriminant signals two unique real solutions.\n- Vertex clarity: The upward-opening parabola and vertex reveal key shape and location.\n- Roots accuracy: Solutions ( x = 1 ) and ( x = 3 ) confirm the validity of algebraic methods.", "---", "### Real-World Applications", "Understanding such equations helps in modeling real-life scenarios—from projectile motion to business profit projections—where quadratic relationships are key.", "---", "### Summary", "For ( a = 3 ), ( b = -12 ), and ( c = 9 ), the quadratic equation:\n- Has real, distinct roots ( x = 1 ) and ( x = 3 ).\n- Opens upwards with vertex at ( (2, -3) ).\n- Has a positive leading coefficient that influences its shape.", "Mastering these concepts strengthens algebraic fluency and prepares learners for more complex mathematical challenges.", "---", "Keywords: quadratic equation, ( a = 3 ), ( b = -12 ), ( c = 9 ), discriminant, roots, vertex, parabola, algebraic solutions, mathematics education.", "---", "Meta Description:\nExplore the quadratic equation ( 3x^2 - 12x + 9 = 0 ) with ( a = 3 ), ( b = -12 ), ( c = 9 ). Learn how discriminant, roots, and graph shape reveal the equation’s solutions and behavior—perfect for students mastering algebra."]