هنا، \(a = 2\)، \(b = -4\)، \(c = -6\)

["Understanding the Quadratic Expression: Here, (a = 2), (b = -4), (c = -6)", "In the study of algebra, particularly in quadratic equations, carefully analyzing the coefficients (a), (b), and (c) is crucial to understanding the behavior of the quadratic function. Today, we explore the specific case where:", "- (a = 2)\n- (b = -4)\n- (c = -6)", "This leads us to the quadratic expression:\n[\nf(x) = 2x^2 - 4x - 6\n]", "### What Is a Quadratic Equation?", "A quadratic equation is any expression of the form:\n[\nf(x) = ax^2 + bx + c\n]\nwhere (a), (b), and (c) are real numbers, and (a <br/>\neq 0). The coefficient (a) determines the parabola’s direction (opens upward if (a > 0), downward if (a < 0)) and its width, while (b) and (c) influence the position of the vertex and the y-intercept.", "---", "### Step 1: Analyzing the Coefficients", "Given (a = 2), (b = -4), and (c = -6):", "- Since (a = 2 > 0), the parabola opens upward, meaning the function has a minimum point (vertex) and not a maximum.\n- The negative (b = -4) indicates the axis of symmetry (the vertical line through the vertex) is located at (x = -\frac{b}{2a}), pointing to the left side of the origin.\n- The negative constant (c = -6) shifts the parabola downward, affecting the y-intercept, where the graph crosses the y-axis at ((0, -6)).", "---", "### Step 2: Finding Key Features of the Quadratic", "#### Vertex (Minimum Point)", "The x-coordinate of the vertex is found using:\n[\nx_v = -\frac{b}{2a} = -\frac{-4}{2 \ imes 2} = \frac{4}{4} = 1\n]", "Substitute (x = 1) into (f(x)) to find the y-coordinate:\n[\nf(1) = 2(1)^2 - 4(1) - 6 = 2 - 4 - 6 = -8\n]", "So, the vertex is at ((1, -8)), the lowest point on the graph.", "#### Y-Intercept", "The y-intercept occurs at (x = 0):\n[\nf(0) = 2(0)^2 - 4(0) - 6 = -6\n]", "---", "### Step 3: Solving (f(x) = 0) — Finding the Roots", "To find the roots, solve:\n[\n2x^2 - 4x - 6 = 0\n]", "Use the quadratic formula:\n[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "Calculate discriminant:\n[\n\Delta = (-4)^2 - 4(2)(-6) = 16 + 48 = 64\n]", "Roots:\n[\nx = \frac{4 \pm \sqrt{64}}{4} = \frac{4 \pm 8}{4}\n]", "So,\n[\nx_1 = \frac{12}{4} = 3, \quad x_2 = \frac{-4}{4} = -1\n]", "The function crosses the x-axis at (x = -1) and (x = 3).", "---", "### Step 4: Behavior and Graph Summary", "- Direction: Upward (due to (a = 2 > 0))\n- Vertex: ((1, -8)) — minimum value\n- Y-intercept: ((0, -6))\n- Roots: (x = -1), (x = 3)\n- Axis of symmetry: (x = 1)\n- Width: Since (|a| = 2), the graph is moderately narrow (wider than (x^2), narrower than (2x^2) with larger (|a|))", "---", "### Practical Applications", "Understanding this quadratic helps in various real-world models such as:", "- Projectile motion (maximum height and trajectory)\n- Cost and revenue analysis (where profit changes parabolically)\n- Physics and engineering design involving acceleration and motion", "---", "### Conclusion", "Analyzing a quadratic with (a = 2), (b = -4), (c = -6) reveals key features: a minimum point at ((1, -8)), roots at (x = -1) and (x = 3), and a downward-parabolical U-shape shifted down by 6 units. Mastering these concepts lays a solid foundation for solving more complex algebraic and applied problems.", "If you're studying quadratics, here are your next steps: practice finding vertex form, graphing with key points, and solving (f(x) = 0) using factoring or the quadratic formula. Confidence in these core skills unlocks deeper understanding in advanced math.", "---", "Keywords: quadratic equation (2x^2 - 4x - 6), roots of quadratic, vertex of parabola, solving quadratic equations, algebra tutorial, quadratic function analysis.\nMeta Description: Explore the quadratic function (f(x) = 2x^2 - 4x - 6), including its vertex, roots, direction, and real-world applications. Perfect for students learning algebra fundamentals."]