Question: Solve for \( b \): If \( b + c = 12 \) and \( b^2 + c^2 = 74 \), find \( b^3 + c^3 \).
["Solving for ( b ) and Calculating ( b^3 + c^3 ): A Step-by-Step Guide", "When faced with equations like ( b + c = 12 ) and ( b^2 + c^2 = 74 ), solving for ( b^3 + c^3 ) requires a smart algebraic approach. This SEO-optimized guide walks you through solving for ( b ) (or related values), then calculating ( b^3 + c^3 ) efficiently using known identities — perfect for students and math enthusiasts seeking clarity and precision.", "---", "### Understanding the Problem", "We are given:", "[\nb + c = 12 \ ag{1}\n]\n[\nb^2 + c^2 = 74 \ ag{2}\n]", "Our goal is to:", "- Solve for ( b ) and ( c ),\n- Compute ( b^3 + c^3 ),", "and present the solution in a search-engine friendly, structured way.", "---", "### Step 1: Use Identities to Simplify Computation", "Instead of solving for ( b ) and ( c ) individually (which involves quadratic solutions), use a useful algebraic identity to compute ( b^3 + c^3 ) directly:", "[\nb^3 + c^3 = (b + c)^3 - 3bc(b + c)\n]", "We already know ( b + c = 12 ), so ( (b + c)^3 = 12^3 = 1728 ).\nBut to apply the identity, we need ( bc ), which we must find.", "---", "### Step 2: Find ( bc ) from the Given Equations", "We use the identity for the square of a sum:", "[\n(b + c)^2 = b^2 + 2bc + c^2\n]", "Substitute known values:", "[\n12^2 = 74 + 2bc\n]\n[\n144 = 74 + 2bc\n]\n[\n2bc = 144 - 74 = 70\n]\n[\nbc = 35\n]", "---", "### Step 3: Plug Into the Cubic Identity", "Now substitute into the formula:", "[\nb^3 + c^3 = (b + c)^3 - 3bc(b + c)\n]\n[\n= 1728 - 3 \cdot 35 \cdot 12\n]\n[\n= 1728 - 1260 = 468\n]", "---", "### Final Answer", "[\n\boxed{b^3 + c^3 = 468}\n]", "---", "### Why This Method Works", "This approach avoids complex solving of quadratic equations by leveraging fundamental algebraic identities:", "- The identity ( b^3 + c^3 = (b + c)^3 - 3bc(b + c) ) rapidly computes the desired value.\n- Finding ( bc ) from ( (b + c)^2 = b^2 + c^2 + 2bc ) is efficient and avoids working with messy factors.", "---", "### SEO Keywords & Phrases Used", "- Solve for ( b )\n- Solve for ( b^3 + c^3 )\n- Algebraic identities\n- Calculate ( b^3 + c^3 )\n- System of equations with ( b ) and ( c )\n- How to find cubic expressions from sum and sum of squares\n- Math problem solution guide", "---", "### Conclusion", "Instead of lengthy substitution or solving quadratics, use identities to efficiently compute ( b^3 + c^3 ) when given ( b + c ) and ( b^2 + c^2 ). This method saves time, reduces error, and strengthens conceptual understanding — essential for mastering algebra in preparation for exams or real-world problem-solving.", "---", "Ready to try your own? Start with: ( b + c = 12 ), ( b^2 + c^2 = 74 ), solve for ( b^3 + c^3 )."]
