Löse nach \( x \): \( 2x^2 - 8x + 6 = 0 \).

["Solving the Quadratic Equation: Löse nach ( x ): ( 2x^2 - 8x + 6 = 0 )", "Quadratic equations are fundamental in algebra, appearing in various fields from physics to engineering. In this article, we’ll explore how to solve the equation ( 2x^2 - 8x + 6 = 0 ) step by step using one of the most effective methods: factoring. Whether you're a student mastering algebra or a learner seeking to strengthen your understanding, this guide will help you solve quadratic equations confidently.", "---", "### Step 1: Understand the Standard Form", "The general form of a quadratic equation is:\n[\nax^2 + bx + c = 0\n]\nIn our case:\n- ( a = 2 )\n- ( b = -8 )\n- ( c = 6 )", "Having the equation in standard form allows us to organize the steps clearly and prepare for factoring.", "---", "### Step 2: Simplify the Equation (If Possible)", "Before factoring, simplify the equation if possible. Here, all coefficients are even, so divide the entire equation by 2 to reduce complexity:\n[\n\frac{2x^2 - 8x + 6}{2} = \frac{0}{2}\n]\n[\nx^2 - 4x + 3 = 0\n]\nNow, the equation is simpler and easier to factor.", "---", "### Step 3: Factor the Quadratic Expression", "We aim to factor the quadratic expression ( x^2 - 4x + 3 ) into two binomials. Look for two numbers ( m ) and ( n ) such that:\n[\nm \cdot n = 3 \quad \ ext{(constant term)}\n]\n[\nm + n = -4 \quad \ ext{(coefficient of } x\ ext{)}\n]", "The pair ( -1 ) and ( -3 ) satisfies both conditions:\n[\n(-1) \cdot (-3) = 3 \quad \ ext{and} \quad -1 + (-3) = -4\n]", "Thus, we can write:\n[\nx^2 - 4x + 3 = (x - 1)(x - 3)\n]", "---", "### Step 4: Solve Using the Zero Product Property", "Set each factor equal to zero and solve for ( x ):\n[\n(x - 1)(x - 3) = 0\n]\nAccording to the zero product property, if ( a \cdot b = 0 ), then ( a = 0 ) or ( b = 0 ).", "So:\n[\nx - 1 = 0 \quad \Rightarrow \quad x = 1\n]\n[\nx - 3 = 0 \quad \Rightarrow \quad x = 3\n]", "---", "### Step 5: Verify the Solutions", "Plug ( x = 1 ) and ( x = 3 ) back into the original equation to confirm:", "- For ( x = 1 ):\n ( 2(1)^2 - 8(1) + 6 = 2 - 8 + 6 = 0 ) ✅", "- For ( x = 3 ):\n ( 2(3)^2 - 8(3) + 6 = 18 - 24 + 6 = 0 ) ✅", "Both solutions satisfy the equation, confirming they are correct.", "---", "### Final Result", "The solutions to the equation ( 2x^2 - 8x + 6 = 0 ) are:\n[\n\boxed{x = 1 \quad \ ext{and} \quad x = 3}\n]", "---", "### Additional Tips for Solving Quadratic Equations", "If factoring proves difficult, alternative methods include:\n- Quadratic Formula: ( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} )\n- Completing the Square: Rewriting the equation into a perfect square trinomial\n- Graphical Solution: Plot the quadratic function and find the x-intercepts", "Mastering these methods expands your ability to tackle more complex algebraic challenges.", "---", "### Why Solving Quadratics Matters", "Understanding how to solve equations like ( 2x^2 - 8x + 6 = 0 ) is essential for modeling real-world phenomena—from projectile motion to profit maximization. Whether you're studying math, science, or engineering, quadratic solutions provide powerful tools for analysis and problem-solving.", "---", "By following these clear steps—simplifying, factoring, solving, and verifying—any quadratic equation becomes manageable. Practice regularly, and soon you’ll identify and solve quadratic equations with confidence!"]