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

["Solving the Quadratic Equation: ( 2x^2 - 8x + 6 = 0 )", "Quadratic equations form a fundamental part of algebra, appearing in both academic curriculums and real-world applications across science, engineering, and economics. One commonly encountered equation is ( 2x^2 - 8x + 6 = 0 ). Solving it not only reinforces core algebraic techniques but also sharpens problem-solving skills. This article guides you through the step-by-step process of solving this quadratic equation, explaining how to find the roots efficiently using both factoring and the quadratic formula.", "---", "### Step-by-Step Solution", "The equation we want to solve is:\n[\n2x^2 - 8x + 6 = 0\n]", "#### 1. Simplify the Equation (Optional but Helpful)\nBefore solving, simplify by dividing all terms by the greatest common divisor (GCD), which is 2:\n[\n\frac{2x^2 - 8x + 6}{2} = \frac{0}{2}\n\Rightarrow x^2 - 4x + 3 = 0\n]\nNow we solve:\n[\nx^2 - 4x + 3 = 0\n]", "#### 2. Factor the Quadratic Expression\nWe now factor the simplified quadratic:\nLook for two numbers that multiply to ( 3 ) (constant term) and add to ( -4 ) (coefficient of ( x )).\nThose numbers are ( -3 ) and ( -1 ), since:\n[\n-3 \ imes (-1) = 3 \quad \ ext{and} \quad -3 + (-1) = -4\n]\nThus, the factored form is:\n[\n(x - 3)(x - 1) = 0\n]", "#### 3. Apply the Zero Product Property\nSet each factor equal to zero:\n[\nx - 3 = 0 \quad \Rightarrow \quad x = 3\n]\n[\nx - 1 = 0 \quad \Rightarrow \quad x = 1\n]", "#### 4. (Optional) Use the Quadratic Formula\nFor completeness, using the quadratic formula:\n[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]\nFor ( x^2 - 4x + 3 = 0 ), coefficients are:\n( a = 1 ), ( b = -4 ), ( c = 3 )\n[\nx = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(1)(3)}}{2(1)} = \frac{4 \pm \sqrt{16 - 12}}{2} = \frac{4 \pm \sqrt{4}}{2} = \frac{4 \pm 2}{2}\n]\nSo:\n[\nx = \frac{4 + 2}{2} = 3 \quad \ ext{and} \quad x = \frac{4 - 2}{2} = 1\n]", "---", "### Final Answer", "The solutions to the equation ( 2x^2 - 8x + 6 = 0 ) are:\n[\n\boxed{x = 1} \quad \ ext{and} \quad \boxed{x = 3}\n]", "---", "### Why This Equation Matters", "Solving quadratic equations like this one builds a strong foundation in algebra. These skills help in modeling parabolic relationships, optimizing functions in economics, analyzing projectile motion in physics, and designing systems in engineering. By mastering both factoring and the quadratic formula, learners gain versatile tools applicable to thousands of real-world problems.", "---", "### Quick Summary: Key Steps Recap\n- Simplify the equation by dividing by common factors\n- Factor to standard quadratic form\n- Apply zero product property or quadratic formula\n- Verify solutions by substitution\n- Recognize real-world relevance", "Learning to solve ( 2x^2 - 8x + 6 = 0 ) isn’t just about finding answers — it’s about developing critical thinking that empowers deeper mathematical and analytical growth.", "---", "Keywords: solve quadratic equation, solve ( 2x^2 - 8x + 6 = 0 ), quadratic formula, factoring method, algebra practice, math tutorial, step-by-step solution, real-world math applications."]