Question: Let $ p $ and $ q $ be real numbers such that $ p + q = 5 $ and $ pq = 6 $. Find the value of $ p^3 + q^3 $.
["Title: Find $ p^3 + q^3 $ Given $ p + q = 5 $ and $ pq = 6 $", "When solving problems involving two variables with known sum and product, algebraic identities offer powerful shortcuts—especially when tasked with finding expressions like $ p^3 + q^3 $. In this article, we’ll explore how to compute $ p^3 + q^3 $ using real numbers $ p $ and $ q $ satisfying the conditions $ p + q = 5 $ and $ pq = 6 $.", "---", "### Understanding the Identity", "The expression $ p^3 + q^3 $ does not immediately factor like a square or a quadratic, but it has a well-known algebraic identity:", "$$\np^3 + q^3 = (p + q)^3 - 3pq(p + q)\n$$", "This identity is derived from expanding $ (p + q)^3 $ and isolating the cubic sum.", "---", "### Plug in Known Values", "We are given:\n- $ p + q = 5 $\n- $ pq = 6 $", "Now substitute into the identity:", "$$\np^3 + q^3 = (5)^3 - 3 \cdot 6 \cdot 5\n$$", "Calculate each term:\n- $ 5^3 = 125 $\n- $ 3 \cdot 6 \cdot 5 = 90 $", "So:", "$$\np^3 + q^3 = 125 - 90 = 35\n$$", "---", "### Verification via Roots of a Quadratic", "For deeper insight, recall that $ p $ and $ q $ are roots of the quadratic equation:", "$$\nx^2 - (p + q)x + pq = 0 \Rightarrow x^2 - 5x + 6 = 0\n$$", "Factoring:", "$$\n(x - 2)(x - 3) = 0 \Rightarrow x = 2 \ ext{ or } 3\n$$", "Thus, $ (p, q) = (2, 3) $ or $ (3, 2) $. Then:", "$$\np^3 + q^3 = 2^3 + 3^3 = 8 + 27 = 35\n$$", "This confirms our earlier result.", "---", "### Why This Matters in SEO", "Optimizing content for search engines involves targeting specific, high-intent query phrases — such as “how to find $ p^3 + q^3 $ given $ p + q $ and $ pq $” — while ensuring technical accuracy and clear explanation. Including both algebraic derivation and verification strengthens content quality, improves readability, and increases chances of ranking for related academic or programming-related searches.", "---", "### Conclusion", "Given $ p + q = 5 $ and $ pq = 6 $, the value of $ p^3 + q^3 $ is 35, efficiently computed using the identity:", "$$\np^3 + q^3 = (p + q)^3 - 3pq(p + q) = 125 - 90 = 35\n$$", "This approach combines mathematical elegance with practical problem-solving—ideal for students, educators, and anyone working with symmetric expressions in algebra."]
