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

["# Solve for ( x ) in the Equation ( 2x^2 - 8x + 6 = 0 ) Using the Quadratic Formula", "Solving quadratic equations is a fundamental skill in algebra, essential for students, teachers, and math enthusiasts alike. One commonly encountered equation is ( 2x^2 - 8x + 6 = 0 ). Whether you're preparing for exams or simply want to strengthen your problem-solving skills, mastering the quadratic formula will equip you to tackle this and similar problems efficiently. In this article, we’ll break down how to solve for ( x ) step by step using the quadratic formula, ensuring clarity and accuracy.", "## Understanding the Quadratic Formula", "The quadratic equation in standard form is:", "[\nax^2 + bx + c = 0\n]", "The quadratic formula provides two solutions for ( x ):", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "This formula works for any quadratic equation and relies on computing the discriminant (( b^2 - 4ac )) to determine the nature of the roots—whether they are real and distinct, real and repeated, or complex.", "## Step 1: Identify Coefficients From the Equation", "Given the equation:", "[\n2x^2 - 8x + 6 = 0\n]", "We identify the coefficients:", "- ( a = 2 ) (coefficient of ( x^2 ))\n- ( b = -8 ) (coefficient of ( x ))\n- ( c = 6 ) (constant term)", "## Step 2: Compute the Discriminant", "The discriminant ( D ) tells us how many real solutions to expect:", "[\nD = b^2 - 4ac = (-8)^2 - 4(2)(6)\n]", "Calculate:", "[\nD = 64 - 48 = 16\n]", "Since ( D = 16 > 0 ), there are two distinct real solutions.", "## Step 3: Apply the Quadratic Formula", "Substitute ( a ), ( b ), and ( D ) into the quadratic formula:", "[\nx = \frac{-(-8) \pm \sqrt{16}}{2 \cdot 2}\n]", "Simplify step by step:", "- Numerator: ( -(-8) = 8 ), so ( 8 \pm \sqrt{16} = 8 \pm 4 )\n- Denominator: ( 2 \cdot 2 = 4 )", "So,", "[\nx = \frac{8 \pm 4}{4}\n]", "This gives two solutions:", "1. ( x = \frac{8 + 4}{4} = \frac{12}{4} = 3 )\n2. ( x = \frac{8 - 4}{4} = \frac{4}{4} = 1 )", "## Step 4: Write the Final Solutions", "The equation ( 2x^2 - 8x + 6 = 0 ) has two real solutions:", "[\nx = 1 \quad \ ext{and} \quad x = 3\n]", "These can also be written as:", "[\nx = 1 \quad \ ext{or} \quad x = 3\n]", "## Why Knowing This Matters", "Solving quadratic equations accurately helps in physics, engineering, economics, and data science applications. Using the quadratic formula ensures a reliable method, especially when factoring is difficult or impossible. Understanding each step builds confidence and deepens algebraic thinking.", "## Summary", "- Start with the standard form ( ax^2 + bx + c = 0 ).\n- Plug coefficients ( a ), ( b ), ( c ) into the quadratic formula.\n- Calculate the discriminant to know the nature of roots.\n- Simplify carefully for accurate solutions.\n- For ( 2x^2 - 8x + 6 = 0 ), the solutions are ( x = 1 ) and ( x = 3 ).", "Mastering these steps ensures you’re equipped to solve any quadratic equation with clarity and precision. Whether you’re studying for a test or solving real-world problems, knowing how to use the quadratic formula is essential.", "---", "Keywords: Solve ( 2x^2 - 8x + 6 = 0 ), quadratic formula, real solutions, discriminant, algebra, step-by-step solution, math tutorial."]