\( r^2 + s^2 + t^2 = (r + s + t)^2 - 2(rs + rt + st) = 6^2 - 2(11) = 36 - 22 = 14 \)

["# Mastering the Identity: ( r^2 + s^2 + t^2 = (r + s + t)^2 - 2(rs + rt + st) = 6^2 - 2(11) = 14 )", "Understanding fundamental algebraic identities is crucial for simplifying complex expressions and solving equations efficiently. One particularly elegant identity is the expansion and rearrangement of ( r^2 + s^2 + t^2 ) in terms of sums and pairwise products. Let’s explore this identity step-by-step and see how it works—with a practical example showing ( r^2 + s^2 + t^2 = 14 ) when ( r + s + t = 6 ) and ( rs + rt + st = 11 ).", "---", "## What Is the Identity ( r^2 + s^2 + t^2 = (r + s + t)^2 - 2(rs + rt + st) )?", "This identity expresses the sum of squares of three variables as the square of their sum minus twice the sum of all pairwise products. It’s derived directly from expanding ( (r + s + t)^2 ):", "[\n(r + s + t)^2 = r^2 + s^2 + t^2 + 2(rs + rt + st)\n]", "Rearranging this equation:", "[\nr^2 + s^2 + t^2 = (r + s + t)^2 - 2(rs + rt + st)\n]", "This simple formula avoids recalculating the square of the sum twice and enables quicker evaluation when values are known.", "---", "### Step-by-Step Computation from the Identity", "Given:\n- ( r + s + t = 6 )\n- ( rs + rt + st = 11 )", "Step 1: Compute ( (r + s + t)^2 ):\n[\n(6)^2 = 36\n]", "Step 2: Compute ( 2(rs + rt + st) ):\n[\n2 \ imes 11 = 22\n]", "Step 3: Subtract to find ( r^2 + s^2 + t^2 ):\n[\nr^2 + s^2 + t^2 = 36 - 22 = 14\n]", "Thus, ( r^2 + s^2 + t^2 = 14 ).", "---", "## Why This Identity Matters", "- Efficiency: It saves computation time when working with symmetric expressions of three variables.\n- Problem Solving: Useful in algebra, statistics, and physics where sums of squares appear in variance and energy calculations.\n- Generalization: Extends naturally to more variables and underpins key formulas in linear algebra and multivariate analysis.", "---", "## Example Real-World Application", "Suppose you’re analyzing three quantities whose combined total is ( 6 = r + s + t ), and their pairwise interactions sum to ( 11 = rs + rt + st ). Saying ( r^2 + s^2 + t^2 = 14 ) lets you immediately summarize how much variability exists across the variables—useful in fields like economics, engineering, or data science.", "---", "## Summary", "The identity\n[\nr^2 + s^2 + t^2 = (r + s + t)^2 - 2(rs + rt + st)\n]\nis a powerful algebraic shortcut rooted in simple expansion. For values ( r + s + t = 6 ) and ( rs + rt + st = 11 ), it yields\n[\nr^2 + s^2 + t^2 = 36 - 22 = 14\n]", "Mastering such identities streamlines complex calculations and deepens conceptual understanding. Whether in classroom exercises or real-world modeling, familiarity with ( r^2 + s^2 + t^2 ) and related forms strengthens analytical skills.", "---", "### Key Takeaways:\n- Use the identity when simplifying sums of squares with known total and pairwise sums.\n- It saves computational effort and highlights hidden structure in equations.\n- Apply this tool across math, science, and engineering disciplines.", "---", "✨ Boost your algebraic fluency. Learn, apply, and Excel with powerful identity shortcuts like this one!"]