Solution: Complete the square for $ x $ and $ y $. For $ x $: $ 4x^2 - 12x = 4(x^2 - 3x) = 4\left[(x - rac{3}{2})^2 - rac{9}{4}

["# Complete the Square for $ x $ and $ y $: A Step-by-Step Guide", "Understanding how to complete the square is a fundamental skill in algebra that unlocks deeper insights into quadratic equations, graphing parabolas, and solving complex mathematical problems. Whether you're working on simplifying expressions, solving equations, or analyzing functions, mastering this technique empowers you with clarity and confidence. In this article, we'll walk through the step-by-step solution of completing the square for the variable $ x $ and discuss how this method extends naturally to $ y $, with real applications and long-term benefits.", "---", "## What Does Completing the Square Mean?", "Completing the square transforms a quadratic expression of the form $ ax^2 + bx + c $ into a perfect square trinomial. This process reveals key properties about the function, including its vertex, axis of symmetry, and minimum or maximum value—essential information in both algebra and calculus.", "For a quadratic in $ x $, completing the square allows us to rewrite:\n$$\n4x^2 - 12x\n$$\nin the form:\n$$\na(x - h)^2 + k,\n$$\nwhich clearly shows the vertex $ (h, k) $ of the parabola.", "---", "## Step-by-Step Solution: Complete the Square for $ x $", "Let’s solve:\n$$\n4x^2 - 12x = 4\left(x^2 - 3x\right)\n$$", "### Step 1: Factor out the coefficient of $ x^2 $", "Since the terms inside the parentheses share $ x^2 $ and $ x $, factor out 4:\n$$\n4\left(x^2 - 3x\right)\n$$", "### Step 2: Complete the square inside the parentheses", "To complete the square for $ x^2 - 3x $, take half the coefficient of $ x $, which is $ -3 $, halve it to get $ -\frac{3}{2} $, and square it:\n$$\n\left(-\frac{3}{2}\right)^2 = \frac{9}{4}\n$$", "Add and subtract $ \frac{9}{4} $ inside the parentheses:\n$$\n4\left(x^2 - 3x + \frac{9}{4} - \frac{9}{4}\right) = 4\left(\left(x - \frac{3}{2}\right)^2 - \frac{9}{4}\right)\n$$", "### Step 3: Distribute and simplify", "$$\n4\left(x - \frac{3}{2}\right)^2 - 4 \cdot \frac{9}{4} = 4\left(x - \frac{3}{2}\right)^2 - 9\n$$", "Thus, the expression $ 4x^2 - 12x $ rewritten by completing the square is:\n$$\n4\left(x - \frac{3}{2}\right)^2 - 9\n$$", "---", "## Extending to $ y $: Same Principle, New Dimensions", "The method of completing the square is equally powerful when applied to expressions involving $ y $. For example, consider $ y^2 + by + c $. By completing the square, we rewrite it as:\n$$\n\left(y + \frac{b}{2}\right)^2 - \left(\frac{b^2}{4} - c\right)\n$$\nwhich reveals the vertex form and helps determine extremum points and symmetry.", "This approach is invaluable in solving quadratic equations in two variables, analyzing quadratic surfaces, and understanding curves in coordinate geometry.", "---", "## Why Completing the Square Matters", "- Solve Quadratic Equations More Easily: Transforms equations into vertex form for immediate solutions.\n- Graph Parabolas Clearly: The vertex form reveals the minimum or maximum point and the direction the parabola opens.\n- Foundation for Advanced Topics: Used in calculus (optimization), geometry, and engineering applications.\n- Generalizable to Multiple Variables: Applies naturally to expressions in $ x $ and $ y $, enriching your analytical toolkit.", "---", "## Practical Example: Finding the Vertex", "Given $ y = 4x^2 - 12x + 5 $, completing the square yields:\n$$\ny = 4\left(x - \frac{3}{2}\right)^2 - 9\n$$\nFrom this, the vertex is at $ \left(\frac{3}{2}, -9\right) $, the parabola opens upward (since coefficient of squared term is positive), and the axis of symmetry is $ x = \frac{3}{2} $.", "---", "## Conclusion", "Completing the square is more than an algebraic technique—it’s a gateway to deeper mathematical understanding and problem-solving agility. By mastering this method for $ x $ and extending it seamlessly to $ y $, you gain clarity in analyzing quadratic functions, interpreting graphs, and tackling real-world mathematical challenges. Whether you're a student, educator, or enthusiast, incorporating completing the square into your algebra toolkit will strengthen your foundation and pave the way for advanced learning.", "Start practicing today—transform your quadratic expressions into powerful, insightful forms with confidence and precision!"]