頂点の \( x \) は \( x = -\frac{b}{2a} = -\frac{12}{2 \times -2} = 3 \) で与えられます。

["頂点の x 座標 roads ( x = -\frac{b}{2a} = -\frac{12}{2 \ imes -2} = 3 ) で أوجدされる | 数学の基礎簡単に解説", "Number patterns and quadratic functions shape much of algebra—understanding the vertex of a parabola is essential for solving equations, graphing, and optimization problems. In this post, we explore how to calculate the x-coordinate of the vertex for a quadratic function, using a concrete example: when ( x = -\frac{b}{2a} = -\frac{12}{2 \ imes -2} = 3 ).", "### What is the Vertex of a Parabola?", "The vertex is the point where the parabola reaches its minimum (if the parabola opens upward) or maximum (if it opens downward). Its x-coordinate determines the axis of symmetry and is key for graphing and solving quadratic equations.", "For a quadratic function in standard form:\n[\nf(x) = ax^2 + bx + c\n]\nThe x-coordinate of the vertex occurs at:\n[\nx = -\frac{b}{2a}\n]\nThis formula comes directly from completing the square or using calculus to find the function's maximum/minimum.", "### Applying the Formula: ( x = -\frac{12}{2 \ imes -2} = 3 )", "Let’s break down the example:", "- The coefficients are: ( a = 2 ), ( b = -12 ), and ( c = \ ext{(unknown, constant term)} ).\n- Plug ( a = 2 ) and ( b = -12 ) into the vertex formula:\n[\nx = -\frac{-12}{2 \ imes 2} = \frac{12}{4} = 3\n]\nSo, the axis of symmetry is the vertical line ( x = 3 ), and the vertex lies somewhere along this line.", "For a figure like ( f(x) = 2x^2 - 12x + c ), the vertex at ( x = 3 ) marks the turning point of the parabola. The actual y-coordinate (the function’s value at ( x = 3 )) depends on ( c ), but the symmetry and key shape are determined by this ( x )-value.", "### Why Is This Formula Important?", "- Graphing: Knowing the vertex quickly places the parabola’s peak or trough on the graph.\n- Optimization: In real-world problems—like maximizing profit or minimizing cost—the vertex often represents the best value.\n- Problem Simplification: Using ( x = -\frac{b}{2a} ) saves time over calculus, making it ideal for algebra and early calculus students.", "### Final Thoughts", "Finding the vertex with ( x = -\frac{b}{2a} = -\frac{12}{2 \ imes -2} = 3 ) is simple but powerful. This calculation reveals the core symmetry of quadratic relationships and empowers students and learners to solve complex equations with confidence. Whether you're graphing, optimizing, or exploring function behavior, mastering this formula is a fundamental stepping stone in mathematics.", "Keywords:\nvertex formula, quadratic function, ( x = -\frac{b}{2a} ), parabola vertex, algebra tips, graphing quadratic, optimize function, mathematics basics", "---", "This approach combines clear explanation, step-by-step math, and practical relevance to help readers understand and apply one of algebra’s most useful concepts. Let me know if you'd like to add visuals or other related topics!"]