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

["Understanding the Equation ( y^2 - 6y = (y - 3)^2 - 9 ): Simplifying Quadratic Expressions Online", "Solving quadratic equations often involves recognizing key algebraic identities and simplifying expressions effectively. One such essential transformation is understanding how to rewrite and simplify the expression ( y^2 - 6y ) into its equivalent form ( (y - 3)^2 - 9 ). This operation not only enhances problem-solving skills but also builds a solid foundation for advanced algebra.", "---", "### Why Rewrite ( y^2 - 6y ) as ( (y - 3)^2 - 9 )?", "At first glance, the expression ( y^2 - 6y ) appears straightforward, but converting it into the completed square form ( (y - 3)^2 - 9 ) reveals important mathematical insights. This transformation helps simplify solving quadratic equations, completing the square, and graphing parabolas.", "---", "### Step-by-Step Derivation", "Start with the original expression:", "[\ny^2 - 6y\n]", "To convert this into a squared binomial minus a constant:", "1. Identify the square component:\n Notice that ( y^2 - 6y ) resembles part of the expansion of ( (y - 3)^2 = y^2 - 6y + 9 ).", "2. Complete the square:\n Add and subtract 9 inside the expression to form the complete square:", "[\n y^2 - 6y = y^2 - 6y + 9 - 9 = (y - 3)^2 - 9\n ]", "This transformation clearly shows how shifting by the constant 9 completes the square.", "---", "### Algebraic Identity Behind the Conversion", "The expression ( y^2 - 6y ) can be written as:", "[\ny^2 - 6y = (y - 3)^2 - 9\n]", "This reveals the identity:", "[\n\boxed{y^2 - 6y = (y - 3)^2 - 9}\n]", "This identity is crucial in simplifying quadratic expressions, especially when solving for roots or analyzing function behavior.", "---", "### Applications of This Identity", "1. Solving Quadratic Equations:\n By rewriting ( y^2 - 6y = k ) as ( (y - 3)^2 - 9 = k ), it becomes easier to isolate the squared term and solve via square roots.", "2. Graphing Functions:\n Expressions in the form ( a(y - h)^2 + k ) describe parabolas. Here, ( (y - 3)^2 - 9 ) indicates a parabola with vertex at ( (3, -9) ).", "3. Completing the Square:\n This transformation is the first step to converting general quadratics into vertex form, which is vital for graphing and optimization.", "---", "### Example: Solving ( y^2 - 6y = 0 )", "Using the identity:", "[\ny^2 - 6y = (y - 3)^2 - 9 = 0\n]", "Rewriting:", "[\n(y - 3)^2 = 9\n]", "Taking square roots:", "[\ny - 3 = \pm 3 \quad \Rightarrow \quad y = 3 + 3 = 6 \quad \ ext{or} \quad y = 3 - 3 = 0\n]", "This confirms the solutions ( y = 0 ) and ( y = 6 ) in a clean, structured way.", "---", "### Conclusion", "The transformation ( y^2 - 6y = (y - 3)^2 - 9 ) is a powerful algebraic tool that simplifies solving and analyzing quadratics. By mastering this identity and completing the square, students enhance their algebraic fluency and deepen their understanding of quadratic functions.", "Whether studying for exams, tackling homework, or exploring advanced mathematics, learning how to express and manipulate quadratic trinomials opens doors to clearer, more effective problem-solving.", "---", "Keywords: quadratic equation, complete the square, algebraic identity, solve quadratics, y² – 6y = (y – 3)² – 9, vertex form, graph quadratic, algebra simplification, solve quadratic equation online, math tutorial algebra, quadratic form derivation."]