5Question: If \( p + q = 10 \) and \( p^2 + q^2 = 58 \), find \( p^3 + q^3 \).
["Title: How to Solve ( p^3 + q^3 ) Given ( p + q = 10 ) and ( p^2 + q^2 = 58 ) – A Step-by-Step Guide", "Understanding how to compute ( p^3 + q^3 ) from simple equations involving sums and sums of squares is a valuable skill in algebra. In this article, we walk through a clear, step-by-step solution to the problem: If ( p + q = 10 ) and ( p^2 + q^2 = 58 ), find ( p^3 + q^3 ).", "---", "### What You’ll Learn", "- How to use algebraic identities to find higher powers from basic symmetric equations.\n- The role of symmetric sums ( p + q ) and ( p^2 + q^2 ) in computing ( p^3 + q^3 ).\n- Practical application of these formulas in contest math and real problem-solving.", "---", "### Step 1: Recall the Identity for ( p^3 + q^3 )", "The key identity in this context is:", "[\np^3 + q^3 = (p + q)^3 - 3pq(p + q)\n]", "If we can determine both ( p + q ) and ( pq ), we can compute ( p^3 + q^3 ) completely.", "Note: We are given ( p + q = 10 ), but we don’t know ( pq ) directly — we only know ( p^2 + q^2 = 58 ).", "---", "### Step 2: Use a Known Identity to Find ( pq )", "We relate the square sum to the product via:", "[\n(p + q)^2 = p^2 + 2pq + q^2\n]", "Substitute the known values:", "[\n10^2 = 58 + 2pq\n]", "[\n100 = 58 + 2pq \quad \Rightarrow \quad 2pq = 42 \quad \Rightarrow \quad pq = 21\n]", "Great! Now we know ( pq = 21 ).", "---", "### Step 3: Plug Values into the ( p^3 + q^3 ) Formula", "Using:", "[\np^3 + q^3 = (p + q)^3 - 3pq(p + q)\n]", "Substitute ( p + q = 10 ), ( pq = 21 ):", "[\np^3 + q^3 = 10^3 - 3 \ imes 21 \ imes 10 = 1000 - 630 = 370\n]", "---", "### Why This Method Works", "- The identity for ( (p+q)^2 ) links ( p + q ) and ( p^2 + q^2 ) directly to ( pq ).\n- Once ( pq ) is known, the cubic sum formula becomes straightforward.\n- This approach avoids guessing or complex factorization — it uses algebra’s power of symmetric expressions.", "---", "### Final Answer", "[\n\boxed{370}\n]", "---", "### Bonus Tips", "- Practice identifying which identity to use based on given sums (like ( p+q ), ( p^2 + q^2 ), or even ( p^3 + q^3 )).\n- Always solve for unknowns (like ( pq )) from lower-degree symmetric identities first.\n- This method applies not just here, but in many Olympiad-style problems — mastery brings clarity and speed.", "---", "Keywords: ( p^3 + q^3 ), symmetric equations, algebraic identities, ( p + q = 10 ), ( p^2 + q^2 = 58 ), how to find ( p^3 + q^3 ), step-by-step algebra, math problem solving", "Meta description: Learn how to find ( p^3 + q^3 ) when given ( p + q = 10 ) and ( p^2 + q^2 = 58 ) using elementary algebra and key identities — ideal for students and math enthusiasts."]
