f(x) = 4x^2 + 12x + 9 = 4(x^2 + 3x) + 9
["# Understanding the Quadratic Function: ( f(x) = 4x^2 + 12x + 9 = 4(x^2 + 3x) + 9 )", "Quadratic functions are foundational in algebra and play a vital role in various mathematical applications, from physics to economics. One particularly elegant and useful form of a quadratic expression is ( f(x) = 4x^2 + 12x + 9 ), which can be rewritten in its completed square form as ( f(x) = 4(x^2 + 3x) + 9 ). This transformation not only simplifies analysis but also enhances comprehension of key features such as vertex, axis of symmetry, and graph behavior.", "## Why Rewrite Quadratics in Completed Square Form?", "Rewriting ( f(x) = 4x^2 + 12x + 9 ) into its factored/simplified form allows us to:", "- Identify the vertex of the parabola directly.\n- Determine the axis of symmetry effortlessly.\n- Evaluate the function’s minimum or maximum value.\n- Solve equations more efficiently.\nFor the given quadratic, the form ( f(x) = 4(x^2 + 3x) + 9 ) enables a straightforward vertex computation using the vertex formula.", "## Step-by-Step Completing the Square", "### Step 1: Factor out the coefficient of ( x^2 )", "Start with the original expression:", "[\nf(x) = 4x^2 + 12x + 9\n]", "Factor out 4 from the first two terms:", "[\nf(x) = 4(x^2 + 3x) + 9\n]", "### Step 2: Complete the square inside the parentheses", "To complete the square for ( x^2 + 3x ), take half the coefficient of ( x ), square it, and add and subtract it inside the parentheses:", "- Half of 3 is ( \frac{3}{2} )\n- Square of ( \frac{3}{2} ) is ( \frac{9}{4} )", "Add and subtract ( \frac{9}{4} ):", "[\nx^2 + 3x + \frac{9}{4} - \frac{9}{4} = \left(x + \frac{3}{2}\right)^2 - \frac{9}{4}\n]", "### Step 3: Substitute back into the function", "Replace inside the parentheses:", "[\nf(x) = 4\left[\left(x + \frac{3}{2}\right)^2 - \frac{9}{4}\right] + 9\n]", "Distribute the 4:", "[\nf(x) = 4\left(x + \frac{3}{2}\right)^2 - 4 \cdot \frac{9}{4} + 9 = 4\left(x + \frac{3}{2}\right)^2 - 9 + 9\n]", "Simplify:", "[\nf(x) = 4\left(x + \frac{3}{2}\right)^2\n]", "## Analyzing the Simplified Form", "The function is now in vertex form:", "[\nf(x) = a(x - h)^2 + k\n]", "Here, ( a = 4 ), ( h = -\frac{3}{2} ), ( k = 0 ).", "- Vertex: The vertex of the parabola is at ( \left( -\frac{3}{2}, 0 \right) ). Since ( a > 0 ), the parabola opens upward, meaning the vertex is the minimum point.\n- Axis of symmetry: A vertical line through the vertex: ( x = -\frac{3}{2} ).\n- Graph Behavior: With ( a = 4 ), the parabola is narrow, indicating a sharp upward curvature.", "## Solving ( f(x) = 4x^2 + 12x + 9 )", "To find the roots, set ( f(x) = 0 ):", "[\n4(x + \frac{3}{2})^2 = 0\n]", "This yields:", "[\nx + \frac{3}{2} = 0 \Rightarrow x = -\frac{3}{2}\n]", "So, the function has a double root at ( x = -\frac{3}{2} ), confirming the vertex lies on the x-axis.", "## Application: Minimizing and Finding Maximums", "Because ( a = 4 > 0 ), the function has a minimum value at the vertex:", "[\nf\left(-\frac{3}{2}\right) = 0\n]", "Thus, the minimum value of ( f(x) ) is 0, achievable only at ( x = -\frac{3}{2} ).", "## Conclusion", "Rewriting ( f(x) = 4x^2 + 12x + 9 ) in completed square form yields deep insights into its graph and behavior. By completing the square, we uncover the vertex at ( \left( -\frac{3}{2}, 0 \right) ), identify symmetry about ( x = -\frac{3}{2} ), and confirm the function’s minimum. This method remains a powerful tool for solving, analyzing, and graphing quadratic functions efficiently.", "Whether optimizing performance in real-world applications or solving academic problems, mastering the technique of completing the square enriches your algebraic toolkit and provides clarity in working with quadratic polynomials.", "---", "Keywords:\nquadratic function, ( f(x) = 4x^2 + 12x + 9 ), vertex form, completing the square, parabola, vertex, analysis, algebra, math tutorial, function graphing"]
