Solve for \(x\) in the equation \(2x^2 - 8x + 6 = 0\).

["How to Solve the Equation (2x^2 - 8x + 6 = 0) – A Step-by-Step Guide", "Solving quadratic equations is a fundamental skill in algebra that applies to various fields, from physics to engineering. In this article, we’ll walk through how to solve the equation (2x^2 - 8x + 6 = 0) step by step, helping you understand the process of finding real (and complex) solutions. Whether you’re a student or a teacher, this guide will clarify the key concepts behind solving quadratic equations.", "---", "### Step-by-Step Solution", "1. Recognize the standard form\nThe standard form of a quadratic equation is:\n[\nax^2 + bx + c = 0\n]\nFor our equation (2x^2 - 8x + 6 = 0), the coefficients are:\n[\na = 2,\quad b = -8,\quad c = 6\n]", "2. Simplify the equation (if possible)\nNotice that all terms are divisible by 2, so we can simplify:\n[\nx^2 - 4x + 3 = 0\n]\nThis simplified form makes solving easier without changing the solutions.", "3. Choose a solving method\nThere are three primary methods to solve quadratics:\n- Factoring\n- Completing the square\n- Quadratic formula", "We’ll use the quadratic formula, which works for any quadratic equation and gives exact solutions:\n[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "4. Plug coefficients into the quadratic formula\nSubstitute (a = 1), (b = -4), and (c = 3):\n[\nx = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(1)(3)}}{2(1)}\n]\n[\nx = \frac{4 \pm \sqrt{16 - 12}}{2}\n]\n[\nx = \frac{4 \pm \sqrt{4}}{2}\n]\n[\nx = \frac{4 \pm 2}{2}\n]", "5. Calculate the two solutions\n[\nx_1 = \frac{4 + 2}{2} = \frac{6}{2} = 3\n]\n[\nx_2 = \frac{4 - 2}{2} = \frac{2}{2} = 1\n]", "---", "### Final Solutions\nThe solutions to (2x^2 - 8x + 6 = 0) are:\n[\nx = 3 \quad \ ext{and} \quad x = 1\n]\nThese represent the x-values where the quadratic function (f(x) = 2x^2 - 8x + 6) intersects the x-axis (the roots of the equation).", "---", "### Why Solving Quadratics Matters\nSolving equations like (2x^2 - 8x + 6 = 0) is essential for modeling real-world phenomena such as projectile motion, optimizing areas, and analyzing electrical circuits. Mastery of these techniques builds a strong foundation for advanced mathematics and engineering applications.", "---", "### Quick Recap: Roots Summary\n- Quadratic Equation: (2x^2 - 8x + 6 = 0)\n- Simplified Form: (x^2 - 4x + 3 = 0)\n- Solutions: (x = 1) and (x = 3)\n- Discriminant: (b^2 - 4ac = 4) (positive, so two distinct real solutions)", "---", "### Alternative Methods (Bonus Tips)\nWhile factoring worked here due to simple coefficients, remember:\n- For prime coefficient quadratics: check factor pairs\n- Completing the square simplifies to vertex form and is useful for graphing\n- The quadratic formula guarantees solutions for all quadratics, even non-factorable or complex ones", "---", "Conclusion\nSolving (2x^2 - 8x + 6 = 0) using the standard quadratic method provides clear, accurate results and reinforces core algebraic skills. Whether you simplify first or apply formulas directly, this problem illustrates how algebraic tools help uncover unknown values efficiently. Practice these steps with other equations to grow confident in mastering quadratics!", "---", "Keywords:\nsolve (2x^2 - 8x + 6 = 0), quadratic equation solution, quadratic formula, algebraic methods, simplify quadratic, real roots, step-by-step solving, algebra tutorial", "Meta Description:\nLearn how to solve (2x^2 - 8x + 6 = 0) with detailed steps using the quadratic formula. Understand factoring, discriminants, and real-world applications in algebra."]