Here, \(a = 2\), \(b = -8\), \(c = 6\).

["Understanding the Quadratic Equation with (a = 2), (b = -8), and (c = 6)", "When analyzing quadratic equations, selecting specific values for coefficients (a), (b), and (c) helps clarify their real-world applications and mathematical behavior. In this article, we focus on the quadratic expression defined by:\n[\nf(x) = 2x^2 - 8x + 6\n]\nwith (a = 2), (b = -8), and (c = 6). This particular equation serves as a classic example in algebra, offering insights into roots, vertex, graph shape, and applications.", "### How to Identify This Quadratic Function", "Given standard form (f(x) = ax^2 + bx + c), substituting the values yields:\n- Leading coefficient (a = 2) → positive, so the parabola opens upwards.\n- Middle coefficient (b = -8) → indicates the axis of symmetry lies to the right of the origin.\n- Constant term (c = 6) → the y-intercept occurs at ((0, 6)).", "These parameters make (f(x) = 2x^2 - 8x + 6) a useful teaching tool to demonstrate key features of quadratic functions.", "---", "### Solving for Roots (Zeros)", "To find where the function crosses the x-axis (the roots), solve:\n[\n2x^2 - 8x + 6 = 0\n]", "Step 1: Simplify the equation\nDivide all terms by 2:\n[\nx^2 - 4x + 3 = 0\n]", "Step 2: Factor the quadratic\nLook for two numbers multiplying to (+3) and adding to (-4):\n[\n(x - 1)(x - 3) = 0\n]", "Step 3: Solve for (x)\nSet each factor equal to zero:\n[\nx - 1 = 0 \Rightarrow x = 1\n]\n[\nx - 3 = 0 \Rightarrow x = 3\n]", "Thus, the roots are (x = 1) and (x = 3), confirming two real, distinct solutions.", "---", "### Calculating the Vertex", "The vertex lies at the axis of symmetry, computed via:\n[\nx = -\frac{b}{2a} = -\frac{-8}{2 \cdot 2} = \frac{8}{4} = 2\n]", "Substitute (x = 2) into the original equation to find the y-coordinate:\n[\nf(2) = 2(2)^2 - 8(2) + 6 = 8 - 16 + 6 = -2\n]", "The vertex is located at ((2, -2)), indicating the minimum point of the upward-opening parabola.", "---", "### Analyzing the Parabola’s Shape and Direction", "Because (a = 2 > 0), the parabola opens upward.\nSince (b) is negative ((-8)), the axis of symmetry (x = 2) lies in the positive x-region, slightly to the right of the origin.", "---", "### Applications and Interpretations", "This quadratic model can represent various real-life scenarios:\n- Projectile Motion: Modeling the height of an object where upward arcs indicate launch trajectories.\n- Profit Optimization: Representing revenue minus costs, where maximum yield (profit) occurs at (x = 2).\n- Geometry: Finding intersections with axes, useful in regression analysis or curve fitting.", "---", "### Summary", "With coefficients (a = 2), (b = -8), and (c = 6), the equation\n[\nf(x) = 2x^2 - 8x + 6\n]\nrepresents a parabola opening upwards with roots at (x = 1) and (x = 3), vertex at ((2, -2)), and practical utility across science and economics. Understanding such equations strengthens foundational algebra skills and supports advanced mathematical modeling.", "Keywords: quadratic equation, (a=2), (b=-8), (c=6), roots, vertex, parabola, algebra, function graph, real roots, axis of symmetry."]