Solve for x: 3x² - 12x + 9 = 0.

["Solve for x: 3x² - 12x + 9 = 0 – Step-by-Step Guide", "Solving quadratic equations is a fundamental skill in algebra, essential for students, educators, and math enthusiasts. One common equation that often comes up is 3x² - 12x + 9 = 0. Whether you're preparing for exams or simply want to master quadratic solutions, this article walks you through solving the equation step-by-step — ensuring clear understanding and confident problem-solving.", "---", "### What is the Equation?", "We are solving:\n3x² - 12x + 9 = 0", "This is a standard quadratic equation in the form:\nax² + bx + c = 0,\nwhere ( a = 3 ), ( b = -12 ), and ( c = 9 ).", "---", "### Step 1: Simplify the Equation", "Before applying the quadratic formula, always check if the equation can be simplified.", "Notice that all coefficients (3, -12, 9) are divisible by 3:", "[\n\frac{3x^2 - 12x + 9}{3} = \frac{0}{3} \implies x^2 - 4x + 3 = 0\n]", "Now we solve:\nx² - 4x + 3 = 0", "---", "### Step 2: Factor the Quadratic (if possible)", "Look for two numbers that multiply to 3 and add to -4.\nThose numbers are -3 and -1, because:\n- (-3) × (-1) = 3\n- (-3) + (-1) = -4", "So the factored form is:\n[\n(x - 3)(x - 1) = 0\n]", "---", "### Step 3: Solve Using Zero Product Property", "Set each factor equal to zero:\n[\nx - 3 = 0 \quad \ ext{or} \quad x - 1 = 0\n]", "Solving gives:\n[\nx = 3 \quad \ ext{or} \quad x = 1\n]", "---", "### Alternative: Applying the Quadratic Formula", "If factoring is difficult, the quadratic formula provides a reliable solution:", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "For original equation 3x² - 12x + 9 = 0,\n- ( a = 3 ),\n- ( b = -12 ),\n- ( c = 9 )", "Plug into the formula:", "[\nx = \frac{-(-12) \pm \sqrt{(-12)^2 - 4(3)(9)}}{2(3)}\n]", "[\nx = \frac{12 \pm \sqrt{144 - 108}}{6} = \frac{12 \pm \sqrt{36}}{6} = \frac{12 \pm 6}{6}\n]", "This gives two solutions:\n[\nx = \frac{12 + 6}{6} = \frac{18}{6} = 3, \quad x = \frac{12 - 6}{6} = \frac{6}{6} = 1\n]", "Both solutions match the factored result: x = 3 and x = 1", "---", "### Why This Equation Matters", "Solving 3x² - 12x + 9 = 0 isn’t just about finding x-values — it teaches key algebraic concepts:\n- Factoring quadratics\n- Applying the quadratic formula\n- Simplifying expressions via greatest common factor\n- Understanding the geometric meaning of roots (x-intercepts of the parabola)", "This equation often appears in real-world modeling, such as maximizing area, projectile motion, or financial break-even analysis.", "---", "### Key Takeaways", "- Divide through by the GCF to simplify when possible.\n- Factoring is efficient when the quadratic is easily factorizable.\n- Always verify solutions by substituting back into the original equation.\n- Understand both factoring and the quadratic formula to handle various quadratic forms.", "---", "### Final Answer", "The solutions to the equation 3x² - 12x + 9 = 0 are:\n[\n\boxed{x = 1 \ ext{ and } x = 3}\n]", "Mastering such equations empowers you to tackle more complex polynomials and strengthens your algebraic foundation. Keep practicing — every quadratic you solve brings you one step closer to algebra expertise!", "---", "Keywords: solve for x, quadratic equation, 3x² - 12x + 9 = 0, factoring, quadratic formula, algebra practice, solve quadratic, math help, intermediate algebra, solve quadratic equation, step-by-step math guide."]