#### 800Question: If $ x + y = 12 $ and $ x^2 + y^2 = 80 $, find $ x^3 + y^3 $.
["How to Solve $ x^3 + y^3 $ Given $ x + y = 12 $ and $ x^2 + y^2 = 80 $", "If you're tackling algebra problems involving sums and powers of variables, one common challenge is calculating expressions like $ x^3 + y^3 $ using only $ x + y $ and $ x^2 + y^2 $. In this article, we’ll solve the problem:\nIf $ x + y = 12 $ and $ x^2 + y^2 = 80 $, find $ x^3 + y^3 $, using efficient algebraic identities.", "---", "### Understanding the Problem", "We’re given:", "- $ x + y = 12 $\n- $ x^2 + y^2 = 80 $", "But we need to find $ x^3 + y^3 $. One of the best tools here is the sum of cubes identity:", "$$\nx^3 + y^3 = (x + y)^3 - 3xy(x + y)\n$$", "This turns our sum $ x + y $ and the sum of squares into a direct computation — as long as we know $ xy $, the computation is straightforward.", "---", "### Step 1: Use Identity Linking $ x^2 + y^2 $ and $ x + y $", "We recall that:", "$$\n(x + y)^2 = x^2 + 2xy + y^2\n$$", "Substitute known values:", "$$\n12^2 = x^2 + y^2 + 2xy\n\quad \Rightarrow \quad\n144 = 80 + 2xy\n$$", "Solving for $ xy $:", "$$\n2xy = 144 - 80 = 64 \quad \Rightarrow \quad xy = 32\n$$", "---", "### Step 2: Plug into the Sum of Cubes Formula", "Now plug $ x + y = 12 $ and $ xy = 32 $ into the identity:", "$$\nx^3 + y^3 = (x + y)^3 - 3xy(x + y)\n= 12^3 - 3 \cdot 32 \cdot 12\n$$", "Calculate each term:", "- $ 12^3 = 1728 $\n- $ 3 \cdot 32 \cdot 12 = 96 \cdot 12 = 1152 $", "So:", "$$\nx^3 + y^3 = 1728 - 1152 = 576\n$$", "---", "### Final Answer", "$$\n\boxed{576}\n$$", "---", "### Why This Method Works", "This approach eliminates the need to solve for $ x $ and $ y $ explicitly. By using algebraic identities, we reduce complex expressions into manageable, computable parts — a powerful strategy in algebra.", "---", "Keywords for SEO:\nx³ + y³ calculator, sum of cubes identity, algebraic equations, solve x³ + y³, x + y and x² + y² problem, step-by-step algebra, solve xy from sum and sum of squares, find x³ + y³ with given values.", "---", "Use this strategy anytime you’re asked to find higher powers from given sum and sum-of-squares!"]
