3x^2 - 12x + 9 = 0 \Rightarrow x^2 - 4x + 3 = 0 \Rightarrow (x-3)(x-1) = 0

["# Solving the Quadratic Equation: 3x² – 12x + 9 = 0 Step-by-Step", "Quadratic equations are fundamental in algebra, appearing frequently in mathematics, science, and engineering. One commonly encountered equation is 3x² – 12x + 9 = 0. This article walks you through solving this equation using algebraic techniques, simplifying it, and applying the fundamental theorem of roots. We’ll show how transforming this equation step-by-step leads to an easy-to-solve quadratic in standard form.", "---", "## Step 1: Start with the Original Equation", "Begin with the quadratic equation:", "[\n3x^2 - 12x + 9 = 0\n]", "This equation models many real-world phenomena like projectile motion, cost optimization, and more. Our goal is to find the values of (x) that satisfy it.", "---", "## Step 2: Simplify by Factoring Out the Greatest Common Factor (GCF)", "Before solving, check if all coefficients share a common factor. Here, the coefficients 3, –12, and 9 share a GCF of 3:", "[\n3(x^2 - 4x + 3) = 0\n]", "Now divide both sides by 3:", "[\nx^2 - 4x + 3 = 0\n]", "This simplification is crucial — it reduces the equation to a simpler, standard quadratic form without altering the solution set.", "---", "## Step 3: Factor the Simplified Quadratic", "Now solve:", "[\nx^2 - 4x + 3 = 0\n]", "Look for two numbers that multiply to (3) (constant term) and add up to (-4) (coefficient of (x)).", "Those numbers are (-3) and (-1), since:", "[\n-3 \ imes -1 = 3 \quad \ ext{and} \quad -3 + (-1) = -4\n]", "Thus, factor the quadratic:", "[\n(x - 3)(x - 1) = 0\n]", "---", "## Step 4: Apply the Zero Product Property", "The zero product property states that if a product of factors is zero, then at least one factor must be zero:", "[\nx - 3 = 0 \quad \ ext{or} \quad x - 1 = 0\n]", "Solving each:", "[\nx = 3 \quad \ ext{or} \quad x = 1\n]", "These are the roots (solutions) of the original equation.", "---", "## Step 5: Final Verification and Summary", "Let’s verify both roots in the original equation:", "- For (x = 3):", "[\n3(3)^2 - 12(3) + 9 = 27 - 36 + 9 = 0 \quad \ ext{✓}\n]", "- For (x = 1):", "[\n3(1)^2 - 12(1) + 9 = 3 - 12 + 9 = 0 \quad \ ext{✓}\n]", "Both values satisfy the equation, confirming correctness.", "---", "## Why This Method Works", "By dividing the original equation by 3, we simplified the quadratic to a monic polynomial (leading coefficient 1), making it easier to factor. Reducing equations step-by-step prevents errors and deepens understanding of algebraic structures — especially factoring, GCF, and root properties.", "---", "## Conclusion", "The equation (3x^2 - 12x + 9 = 0) simplifies to the more intuitive (x^2 - 4x + 3 = 0), which factors neatly into ((x-3)(x-1) = 0). Applying the zero product property gives the solutions (x = 1) and (x = 3). This clean process not only yields the solution but strengthens foundational problem-solving skills in algebra.", "---", "### Related Keywords for SEO:\n- How to solve 3x² – 12x + 9 = 0\n- Quadratic equations step-by-step\n- Factoring quadratic equations\n- Solving x² – 4x + 3 = 0\n- Finding roots using factoring\n- Algebraic solutions for quadratic equations", "---", "Ready to master quadratics? Start by simplifying equations like this one — each step builds a strong foundation in algebra!"]