\[ 12x^2 - 18x + 6 = 0. \]

["# Solving the Quadratic Equation: 12x² – 18x + 6 = 0", "Understanding how to solve quadratic equations is essential in algebra and plays a vital role in many scientific and engineering applications. One such equation is 12x² – 18x + 6 = 0, a classic quadratic that can be efficiently solved using various methods. In this article, we’ll explore step-by-step solving techniques, discuss the nature of the roots, and highlight real-world applications.", "## What Is the Equation?", "The equation you’re analyzing is:\n12x² – 18x + 6 = 0", "This is a standard quadratic equation in the form ax² + bx + c = 0, where:\n- a = 12\n- b = –18\n- c = 6", "Quadratic equations have the potential to yield up to two real solutions or complex roots, depending on the discriminant, calculated as:\n[ \Delta = b^2 - 4ac ]", "## Step 1: Simplify the Equation", "Before solving, it’s helpful to simplify the equation by dividing all terms by the greatest common divisor (GCD) of the coefficients. Here, the GCD of 12, 18, and 6 is 6.", "[ \frac{12x² – 18x + 6}{6} = 0 \Rightarrow 2x² – 3x + 1 = 0 ]", "Now the equation becomes simpler:\n2x² – 3x + 1 = 0", "## Step 2: Determine the Nature of Roots via Discriminant", "Calculate the discriminant to understand the nature of the roots:\n[ \Delta = b² - 4ac = (-3)² - 4(2)(1) = 9 – 8 = 1 ]", "Since Δ = 1 > 0, the equation has two distinct real roots.", "## Step 3: Solve Using Factoring", "With the simplified equation 2x² – 3x + 1 = 0, we attempt factoring:", "We look for two numbers that multiply to ( 2 \ imes 1 = 2 ) and add up to –3. These numbers are –1 and –2.", "Rewrite the middle term:\n[ 2x² – 2x – x + 1 = 0 ]\nGroup terms:\n[ (2x² – 2x) – (x – 1) = 0 ]\nFactor each group:\n[ 2x(x – 1) – 1(x – 1) = 0 ]\nFactor out (x – 1):\n[ (2x – 1)(x – 1) = 0 ]", "## Step 4: Apply Zero Product Property", "Set each factor equal to zero:\n- ( 2x – 1 = 0 \Rightarrow x = \frac{1}{2} )\n- ( x – 1 = 0 \Rightarrow x = 1 )", "## Step 5: Final Solutions", "The solutions to the equation 12x² – 18x + 6 = 0 are:\n[ \boxed{x = \frac{1}{2}} \quad \ ext{and} \quad \boxed{x = 1} ]", "## Applications of Solving Quadratic Equations", "Quadratic equations model real-world phenomena such as:\n- Projectile motion in physics\n- Optimization problems in economics\n- Shape and area calculations\n- Electrical current relationships", "Mastering their solution equips you with tools to analyze and compute in diverse scientific and technical fields.", "## Summary: Key Takeaways\n- Always simplify equations by dividing by GCD when possible.\n- Use the discriminant to determine root nature.\n- Factoring is efficient when the quadratic can be broken down with integer roots.\n- Verifying solutions by plugging them back into the original equation helps confirm accuracy.", "Understanding how to solve 12x² – 18x + 6 = 0 lays a strong foundation for tackling more complex algebraic challenges. Whether you’re a student, educator, or lifelong learner, quadratic equations remain fundamental to progress in STEM disciplines.", "---", "Keywords: quadratic equation, solve 12x² – 18x + 6 = 0, method 2x² – 3x + 1 = 0, discriminant, factoring quadratic, algebra solution, real roots, simplified quadratic equation."]