For $ y $: $ -4(y^2 - 6y) = -4[(y - 3)^2 - 9] $.

["Solving the Equation: $ -4(y^2 - 6y) = -4[(y - 3)^2 - 9] $ – Step-by-Step Guide", "Understanding how to simplify and solve equations like $ -4(y^2 - 6y) = -4[(y - 3)^2 - 9] $ is essential for mastering algebra and developing stronger problem-solving skills. This comprehensive guide breaks down the equation systematically, providing clear explanations and insightful steps to reveal why both sides of the equation are true.", "---", "### Understanding the Equation Structure", "At first glance, the equation:\n$$\n-4(y^2 - 6y) = -4[(y - 3)^2 - 9]\n$$\nseems complex, but it hides a powerful algebraic identity. Both sides contain expressions involving quadratic terms, and the right-hand side explicitly shows a completed square form. Let’s explore how we arrive at this simplified form and verify the equivalence.", "---", "### Step 1: Expand the Expressions", "Start by expanding the left-hand side:\n$$\n-4(y^2 - 6y) = -4y^2 + 24y\n$$", "Now work on the right-hand side. First expand $ (y - 3)^2 $:\n$$\n(y - 3)^2 = y^2 - 6y + 9\n$$\nThen\n$$\n(y - 3)^2 - 9 = y^2 - 6y + 9 - 9 = y^2 - 6y\n$$", "Now multiply by $-4$:\n$$\n-4[(y - 3)^2 - 9] = -4(y^2 - 6y) = -4y^2 + 24y\n$$", "---", "### Step 2: Compare Both Sides", "We now have:\n- Left-hand side: $ -4y^2 + 24y $\n- Right-hand side: $ -4y^2 + 24y $", "Since both expressions are identical, the original equation simplifies to:\n$$\n-4y^2 + 24y = -4y^2 + 24y\n$$", "This confirms the equation is an identity, meaning it holds true for all real values of $ y $. The presence of equal quadratic terms and equal linear coefficients explains the equivalence.", "---", "### Step 3: Why $ (y - 3)^2 - 9 $?", "Expanding $ (y - 3)^2 - 9 $ helps rewrite the expression in completed square form. While it didn’t alter the final form in this case, it emphasizes a shift:\n$$\n(y - 3)^2 - 9 = y^2 - 6y + 9 - 9 = y^2 - 6y\n$$", "This is exactly $ y^2 - 6y $, which matches the expression inside the parentheses on the left. Multiplying back by $-4$ perfectly aligns both sides, reducing the equation to a simple true statement.", "---", "### Step 4: Solving the Equation — What Can We Do?", "Since the equation simplifies to an identity, it does not have discrete solutions—it is true for all real numbers $ y $. However, recognizing this structure empowers us to rewrite complex forms more elegantly, especially useful in calculus, graphing, and higher-level math.", "If rewriting had been required to solve, we’d simplify to:\n$$\n-4(y^2 - 6y) = -4[(y - 3)^2 - 9]\n\Rightarrow (y - 3)^2 - 9 = y^2 - 6y\n\Rightarrow \ ext{True for all } y\n$$", "---", "### Step 5: Key Takeaways", "- Identities preserve equality for all values.\n- Factoring and completing the square simplify expressions and reveal underlying structure.\n- Expanding and comparing both sides confirms equivalence.\n- Recognizing $ (y - 3)^2 - 9 $ clarifies transformations and equality in quadratic forms.", "---", "### Final Thoughts", "Mastering equations like this deepens algebraic intuition and prepares students for advanced mathematics, including derivatives, integrals, and modeling real-world phenomena. Understanding why $ -4(y^2 - 6y) = -4[(y - 3)^2 - 9] $ isn’t just about solving—it’s about recognizing elegant patterns in algebra.", "If you’re studying quadratic expressions or identities, practicing forms like these strengthens your ability to manipulate and interpret equations confidently.", "---", "Keywords:\nMath homework help, solving quadratic equations, algebra identity, completing the square, algebraic simplification, خطوة بخطوة معادلة جبرية, $ -4(y^2 - 6y) $, $ -4[(y - 3)^2 - 9] $, step-by-step algebra, how to solve equations, identity verification, algebraic structure, mathematical reasoning.", "---", "Try it yourself: Plug in any real number for $ y $—the equation will always be true. That’s the beauty and power of algebraic identity!"]