x = rac{-(-6) \pm \sqrt{(-6)^2 - 4 \cdot 3 \cdot 2}}{2 \cdot 3} = rac{6 \pm \sqrt{36 - 24}}{6} = rac{6 \pm \sqrt{12}}{6}

["Solving Quadratic Equations: A Step-by-Step Guide to the Quadratic Formula", "Solving quadratic equations is a fundamental skill in algebra, and understanding the quadratic formula provides a powerful tool for finding exact solutions. One common method involves applying the formula directly to standard quadratic expressions. In this article, we’ll explore how to solve the equation\n[\nx = \frac{-(-6) \pm \sqrt{(-6)^2 - 4 \cdot 3 \cdot 2}}{2 \cdot 3}\n]\nby breaking down each step clearly and explaining the underlying math.", "---", "### Understanding the Quadratic Formula", "The quadratic formula solves equations of the form:\n[\nax^2 + bx + c = 0\n]\nThe solution is:\n[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]\nWhere:\n- (a), (b), and (c) are coefficients from the quadratic equation.\n- The expression under the square root, (b^2 - 4ac), is called the discriminant — it determines the nature of the roots (real and distinct, real and repeated, or complex).\n- The denominator (2a) scales the entire expression.", "---", "### Step 1: Rewrite the Given Equation", "The equation we start with is:\n[\nx = \frac{-(-6) \pm \sqrt{(-6)^2 - 4 \cdot 3 \cdot 2}}{2 \cdot 3}\n]", "Simplify the numerator:\n(-(-6) = +6), so the equation becomes:\n[\nx = \frac{6 \pm \sqrt{(-6)^2 - 4 \cdot 3 \cdot 2}}{2 \cdot 3}\n]", "---", "### Step 2: Compute the Discriminant", "Calculate each part under the square root:\n- ( (-6)^2 = 36 )\n- ( 4 \cdot 3 \cdot 2 = 24 )\n- Discriminant: ( 36 - 24 = 12 )", "So now the equation becomes:\n[\nx = \frac{6 \pm \sqrt{12}}{6}\n]", "---", "### Step 3: Simplify the Square Root", "The expression (\sqrt{12}) can be simplified since 12 is not a perfect square but has a square factor:\n[\n\sqrt{12} = \sqrt{4 \cdot 3} = \sqrt{4} \cdot \sqrt{3} = 2\sqrt{3}\n]", "Substitute back:\n[\nx = \frac{6 \pm 2\sqrt{3}}{6}\n]", "---", "### Step 4: Simplify the Fraction", "Factor numerator and denominator:\n[\nx = \frac{2(3 \pm \sqrt{3})}{2 \cdot 3} = \frac{3 \pm \sqrt{3}}{3}\n]", "So the final simplified solutions are:\n[\nx = \frac{3 + \sqrt{3}}{3} \quad \ ext{and} \quad x = \frac{3 - \sqrt{3}}{3}\n]", "---", "### Why Simplify?", "Keeping answers in simplest form improves readability and usability, especially when these solutions are used in further calculations or graphical representations. While (\frac{6 \pm \sqrt{12}}{6}) is mathematically correct, reducing it to (\frac{6 \pm 2\sqrt{3}}{6}) (and then to (\frac{3 \pm \sqrt{3}}{3})) is clearer and streamlined.", "---", "### Real-World Applications", "Quadratic equations appear in physics (projectile motion), engineering (structural analysis), economics (profit maximization), and many other fields. Mastering the quadratic formula — including efficient simplification — enables accurate modeling and problem-solving across disciplines.", "---", "### Conclusion", "Solving quadratic equations using the formula is straightforward once you break down each component: identify coefficients, compute the discriminant, handle radicals carefully, and simplify fully. In this example, starting with\n[\nx = \frac{-(-6) \pm \sqrt{(-6)^2 - 4 \cdot 3 \cdot 2}}{2 \cdot 3}\n]\nled neatly to the simplified solutions:\n[\n\boxed{x = \frac{3 \pm \sqrt{3}}{3}}\n]\nWith practice, applying the quadratic formula becomes intuitive and efficient.", "---", "Keywords: quadratic formula, solve quadratic equation, discriminant, simplify radical, x = [±√D]/(2a), algebraic equations, step-by-step algebra, trig quadratic simplification\nMeta description: Learn how to solve x = −(−6)±√[(−6)²−4·3·2]/(2·3) step-by-step using the quadratic formula, simplifying to final answers with real-world applications."]