Solution: We are given $ a + b = 10 $ and $ a^2 + b^2 = 58 $. We want to find $ a^3 + b^3 $.
["Solving for a³ + b³ Given a + b = 10 and a² + b² = 58", "When faced with an algebraic expression like ( a^3 + b^3 ), and given the constraints ( a + b = 10 ) and ( a^2 + b^2 = 58 ), many students and learners wonder how to approach the problem efficiently. Fortunately, with smart algebraic identities, we can solve this quickly without unnecessary complexity.", "### Understanding the Problem", "We are given:", "- ( a + b = 10 )\n- ( a^2 + b^2 = 58 )", "We aim to compute:", "- ( a^3 + b^3 )", "Rather than guessing or substituting values blindly, we use well-known formulas to unlock this efficiently.", "---", "### Step 1: Use the Identity for Sum of Cubes", "The formula for the sum of cubes is:", "[\na^3 + b^3 = (a + b)^3 - 3ab(a + b)\n]", "We already know ( a + b = 10 ), so compute ( (a + b)^3 ):", "[\n(10)^3 = 1000\n]", "So now:", "[\na^3 + b^3 = 1000 - 3ab \cdot 10 = 1000 - 30ab\n]", "Thus, to find ( a^3 + b^3 ), we need to determine ( ab ).", "---", "### Step 2: Find ( ab ) from Given Data", "We use another identity that connects ( a + b ), ( a^2 + b^2 ), and ( ab ):", "[\n(a + b)^2 = a^2 + 2ab + b^2\n]", "Substitute known values:", "[\n10^2 = 58 + 2ab\n]\n[\n100 = 58 + 2ab\n]\n[\n2ab = 100 - 58 = 42\n]\n[\nab = 21\n]", "---", "### Step 3: Plug Back into Sum of Cubes Formula", "Now that we know ( ab = 21 ), substitute into the expression:", "[\na^3 + b^3 = 1000 - 30 \cdot 21 = 1000 - 630 = 370\n]", "---", "### Final Answer", "[\n\boxed{a^3 + b^3 = 370}\n]", "---", "### Bonus Tips for Similar Problems", "- Always use identities like ( (a + b)^2 = a^2 + 2ab + b^2 ) to find missing products.\n- Remember the sum of cubes and squares to cubic expressions formulas.\n- This method reduces computational effort and avoids solving quadratic equations unnecessarily.", "Using structured algebraic thinking, complex-looking problems become simple and solvable step-by-step—perfect for mastering algebra efficiently."]
