For the equation \(3x^2 - 12x + 9 = 0\), the coefficients are \(a = 3\), \(b = -12\), and \(c = 9\).
["# Solving the Quadratic Equation (3x^2 - 12x + 9 = 0): A Complete Guide", "When faced with a quadratic equation like (3x^2 - 12x + 9 = 0), correctly identifying the coefficients is essential for applying efficient solving methods. Understanding the structure of a quadratic equation in standard form, (ax^2 + bx + c = 0), allows students and math enthusiasts to confidently use techniques such as factoring, completing the square, or the quadratic formula.", "## Understanding the Coefficients in the Equation", "In the equation (3x^2 - 12x + 9 = 0), the coefficients are clearly defined based on the standard quadratic form:", "- (a = 3)\n This is the coefficient of (x^2). It determines the parabola’s width and direction — positive values open upward, negative values open downward.", "- (b = -12)\n This is the linear coefficient, governing the slope of the parabola’s vertex and influencing the location of the roots.", "- (c = 9)\n This is the constant term, the y-intercept of the parabola when (x = 0).", "With (a = 3), (b = -12), and (c = 9), we are equipped to solve this equation using multiple algebraic approaches.", "## Solving (3x^2 - 12x + 9 = 0) Step-by-Step", "### Step 1: Simplify the Equation (if possible)", "Since all terms are divisible by 3, the equation can be simplified:", "[\n3x^2 - 12x + 9 = 0 \quad \Rightarrow \quad x^2 - 4x + 3 = 0\n]", "Now the simpler equation (x^2 - 4x + 3 = 0) is easier to factor.", "### Step 2: Factor the Quadratic Expression", "We seek two numbers that multiply to (c/a = 1 \ imes 3 = 3) and add to (b/a = -4). These numbers are (-3) and (-1):", "[\nx^2 - 4x + 3 = (x - 3)(x - 1) = 0\n]", "### Step 3: Solve for (x)", "Set each factor equal to zero:", "[\nx - 3 = 0 \quad \Rightarrow \quad x = 3\n]\n[\nx - 1 = 0 \quad \Rightarrow \quad x = 1\n]", "### Step 4: Verify Using the Quadratic Formula", "For completeness, use the quadratic formula:", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "Here, (a = 1), (b = -4), (c = 3) (from simplified form):", "[\nx = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(1)(3)}}{2(1)} = \frac{4 \pm \sqrt{16 - 12}}{2} = \frac{4 \pm \sqrt{4}}{2} = \frac{4 \pm 2}{2}\n]", "So,", "[\nx = \frac{4 + 2}{2} = 3 \quad \ ext{and} \quad x = \frac{4 - 2}{2} = 1\n]", "Both methods confirm the roots are (x = 1) and (x = 3).", "## Why Identifying Coefficients Matters", "Recognizing (a), (b), and (c) allows for quick assessment of:", "- The equation’s concavity and vertex shape\n- Efficiency in choosing solution methods — factoring, formula, or completing the square\n- Understanding the relationship between roots and coefficients (e.g., sum and product)", "## Final Answer", "The solutions to the equation (3x^2 - 12x + 9 = 0) are:", "[\n\boxed{x = 1 \quad \ ext{and} \quad x = 3}\n]", "These values satisfy the original quadratic, confirming the correct resolution using the full coefficient analysis.", "---", "Keywords: quadratic equation (3x^2 - 12x + 9 = 0), coefficients (a), (b), (c), solving quadratic, factoring, quadratic formula, algebraic methods, roots of a quadratic, intermediate algebra."]
