\( S_{n-1} = 3(n-1)^2 + 5(n-1) = 3(n^2 - 2n + 1) + 5n - 5 = 3n^2 - 6n + 3 + 5n - 5 \)
["### Unlocking the Quadratic Formula: Expanding and Simplifying ( S_{n-1} = 3(n-1)^2 + 5(n-1) )", "Understanding how quadratic expressions simplify and expand can make solving problems involving sequences, summations, and polynomial identities much easier. One powerful example is the expression\n[ S_{n-1} = 3(n-1)^2 + 5(n-1) ]\nThis formula, often seen in discrete mathematics or sequence analysis, represents a quadratic in ( n ). In this article, we break down its expansion and exploration step-by-step, helping you simplify and manipulate such expressions with confidence.", "---", "### Step 1: Expand ( (n-1)^2 )", "Start by expanding the squared term:\n[ (n - 1)^2 = n^2 - 2n + 1 ]", "Substitute into ( S_{n-1} ):\n[\nS_{n-1} = 3(n^2 - 2n + 1) + 5(n - 1)\n]", "---", "### Step 2: Distribute the Constants", "Now distribute the coefficients:\n[\n3(n^2 - 2n + 1) = 3n^2 - 6n + 3\n]\n[\n5(n - 1) = 5n - 5\n]", "Add both results:\n[\nS_{n-1} = (3n^2 - 6n + 3) + (5n - 5)\n]", "---", "### Step 3: Combine Like Terms", "Group and combine the terms:\n- ( 3n^2 ) remains unchanged (no other ( n^2 ) terms)\n- ( -6n + 5n = -n )\n- ( 3 - 5 = -2 )", "So the simplified form is:\n[\nS_{n-1} = 3n^2 - n - 2\n]", "---", "### Why This Simplification Matters", "Expressions like ( S_{n-1} ) often appear when evaluating terms in sequences, particularly in recursive relations or summation formulas. The simplified form:\n[\nS_{n-1} = 3n^2 - n - 2\n]\nis easier to plug into summation tools, compute sums, or compare against general formulas. It also reveals the quadratic nature of the sequence — key for finding closed-form expressions or analyzing growth rates.", "---", "### Applications and Next Steps", "- Summation Analysis: When calculating ( \sum S_k ) from ( k = 1 ) to ( n-1 ), the simplified quadratic form makes it straightforward to apply summation rules and generate closed-form expressions.\n- Roots and Behavior: Finding the discriminant of ( 3n^2 - n - 2 ) helps determine whether roots are real, complex, or rational — useful in algorithm complexity or recurrence solving.\n- Generalization: The method used here (expand, distribute, combine) applies to most quadratic models in discrete math.", "---", "### Summary", "Simplifying expressions like\n[ S_{n-1} = 3(n-1)^2 + 5(n-1) ]\nfrom expanded form to ( 3n^2 - n - 2 ) reveals clarity and utility. Mastering this algebraic technique opens doors to deeper problem-solving in sequences, series, and discrete math. Start practicing today — expand, simplify, and inspire confident math mastery!", "---", "If you're studying sequences or preparing for advanced algebra, mastering quadratic expansion and simplification is a vital skill. Remember:\n[\n3(n-1)^2 + 5(n-1) = 3n^2 - n - 2\n]\nis your key to unlocking a broader class of polynomials and their implications!", "---", "Keywords:\nquadratic expansion, polynomial simplification, ( S_{n-1} = 3(n-1)^2 + 5(n-1) ), algebraic simplification, discrete mathematics, summation formulas, generating quadratic forms, simplify polynomial, algebra tip."]
