Question:** 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", "Quadratic equations are fundamental in algebra and appear frequently in science, engineering, and everyday problem-solving. One common challenge students face is solving equations of the form (ax^2 + bx + c = 0). The quadratic formula offers a reliable and universal method for finding solutions, regardless of how complicated the coefficients look. In this article, we’ll walk through solving the equation:", "[\n2x^2 - 8x + 6 = 0\n]", "using the quadratic formula step by step.", "## Why Use the Quadratic Formula?", "The quadratic formula is derived from completing the square and provides direct solutions to any quadratic equation:", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "This method works for all quadratic equations, even when factoring is difficult or impossible.", "## Step 1: Identify coefficients (a), (b), and (c)", "Rewriting the equation (2x^2 - 8x + 6 = 0), we identify:", "[\na = 2,\quad b = -8,\quad c = 6\n]", "## Step 2: Compute the Discriminant", "Before applying the formula, calculate the discriminant (D = b^2 - 4ac) to determine the nature of the solutions.", "[\nD = (-8)^2 - 4(2)(6) = 64 - 48 = 16\n]", "Since (D = 16 > 0), there are two distinct real solutions.", "## Step 3: Apply the Quadratic Formula", "Plug (a), (b), and (c) into the formula:", "[\nx = \frac{-(-8) \pm \sqrt{16}}{2 \cdot 2} = \frac{8 \pm 4}{4}\n]", "Now compute both possible solutions:", "[\nx_1 = \frac{8 + 4}{4} = \frac{12}{4} = 3\n]", "[\nx_2 = \frac{8 - 4}{4} = \frac{4}{4} = 1\n]", "## Step 4: Final Answer", "The solutions to the equation (2x^2 - 8x + 6 = 0) are:", "[\nx = 1 \quad \ ext{and} \quad x = 3\n]", "Thus, the roots are (x = 1) and (x = 3), meaning the quadratic crosses the x-axis at these points.", "## Summary", "- The equation (2x^2 - 8x + 6 = 0) is solved using the quadratic formula.\n- The discriminant confirms two real solutions.\n- Solutions are (x = 1) and (x = 3).", "Mastering the quadratic formula helps you confidently tackle any quadratic equation—whether in homework or real-world applications. Practice helps reinforce this essential algebraic skill!", "---", "Keywords: solve (2x^2 - 8x + 6 = 0), quadratic formula, algebraic solutions, quadratic equations, real roots, discriminant, step-by-step quadratic, solve (x^2 = 1), quadratic formula tutorial, real number solutions.", "---", "Understanding how to solve quadratic equations using the quadratic formula empowers you with a versatile tool. Remember: identify (a), (b), (c); compute the discriminant; apply the formula; and interpret the results. With practice, you’ll solve quadratic equations with ease."]
