Question: If $ x + y = 10 $ and $ x^2 + y^2 = 58 $, find $ x^3 + y^3 $.
["# How to Find $ x^3 + y^3 $ Given $ x + y = 10 $ and $ x^2 + y^2 = 58 $", "When faced with a math problem like If $ x + y = 10 $ and $ x^2 + y^2 = 58 $, find $ x^3 + y^3 $, it might seem difficult at first—but with the right algebraic approach, it becomes straightforward. This article will guide you step-by-step on how to solve for $ x^3 + y^3 $ using fundamental algebraic identities.", "---", "## Understanding the Problem", "We are given:\n- $ x + y = 10 $\n- $ x^2 + y^2 = 58 $", "We need to find:\n- $ x^3 + y^3 $", "Rather than solving explicitly for $ x $ and $ y $, which involves messy quadratics, we use known identities to compute $ x^3 + y^3 $ more efficiently.", "---", "## Key Algebraic Identity", "One of the most powerful identities in symmetric algebra is:", "$$\nx^3 + y^3 = (x + y)^3 - 3xy(x + y)\n$$", "This identity allows us to express the sum of cubes directly in terms of $ x + y $ and $ xy $, which we can calculate using the given data.", "---", "## Step 1: Use $ x + y = 10 $ in the identity", "First, compute $ (x + y)^3 $:", "$$\n(x + y)^3 = 10^3 = 1000\n$$", "Now, substitute into the identity:", "$$\nx^3 + y^3 = 1000 - 3xy(10) = 1000 - 30xy\n$$", "So, to finish, we need the value of $ xy $, which we can find from $ x^2 + y^2 $.", "---", "## Step 2: Compute $ xy $ using $ x^2 + y^2 = 58 $", "We use the identity:", "$$\n(x + y)^2 = x^2 + 2xy + y^2\n$$", "Substitute known values:", "$$\n10^2 = 58 + 2xy\n$$\n$$\n100 = 58 + 2xy\n$$\n$$\n2xy = 100 - 58 = 42\n$$\n$$\nxy = 21\n$$", "---", "## Step 3: Plug $ xy = 21 $ back into the expression for $ x^3 + y^3 $", "$$\nx^3 + y^3 = 1000 - 30 \cdot 21 = 1000 - 630 = 370\n$$", "---", "## Final Answer", "$$\n\boxed{x^3 + y^3 = 370}\n$$", "---", "## Why This Method Works", "By combining $ x + y $, $ x^2 + y^2 $, and the algebraic identity for $ x^3 + y^3 $, we avoided deriving quadratic equations for $ x $ and $ y $. This approach uses symmetry and known formulas to simplify the problem—ideal for math students and enthusiasts tackling structured algebra problems.", "---", "## Related Keywords for SEO Optimization", "- Solving $ x^3 + y^3 $ given $ x + y $ and $ x^2 + y^2 $\n- Algebraic identity $ x^3 + y^3 = (x + y)^3 - 3xy(x + y) $\n- How to find $ x^3 + y^3 $ when $ x + y $ and $ x^2 + y^2 $ are known\n- Step-by-step solution to symmetric equations in algebra\n- Math problem: Given $ x + y = 10 $, $ x^2 + y^2 = 58 $, find $ x^3 + y^3 $", "---", "Optimizing your article with these keywords ensures visibility for learners seeking clear, accurate algebra guidance. Focus on clarity, correctness, and relatability—perfect for math students and educators alike."]
