Question: Let $ p $ and $ q $ be complex numbers such that $ p + q = 3 + 4i $ and $ p^2 + q^2 = 10 - 24i $. Find $ pq $.
["Title: How to Find $ pq $ Given $ p + q $ and $ p^2 + q^2 $: A Complex Number Problem Solved", "Complex number problems often appear in advanced mathematics, engineering, and physics, requiring familiarity with algebraic identities and complex arithmetic. This article explores a classic application: given the sum and sum of squares of two complex numbers $ p $ and $ q $, determine their product $ pq $. We’ll solve a problem where $ p + q = 3 + 4i $ and $ p^2 + q^2 = 10 - 24i $, revealing $ pq $ through elegant algebraic reasoning.", "## The Foundation: Identity Linking Sum, Sum of Squares, and Product", "For any two numbers $ p $ and $ q $, the square of their sum relates to their individual squares via:\n$$\n(p + q)^2 = p^2 + q^2 + 2pq\n$$\nRewriting this to solve for $ pq $:\n$$\npq = \frac{(p + q)^2 - (p^2 + q^2)}{2}\n$$\nThis identity works for complex numbers just like for real ones.", "## Step 1: Compute $ (p + q)^2 $", "Given $ p + q = 3 + 4i $, compute its square:\n$$\n(3 + 4i)^2 = 3^2 + 2 \cdot 3 \cdot 4i + (4i)^2 = 9 + 24i + 16i^2\n$$\nSince $ i^2 = -1 $,\n$$\n= 9 + 24i - 16 = -7 + 24i\n$$", "## Step 2: Plug Into the Identity", "Now substitute $ (p + q)^2 $ and $ p^2 + q^2 $ into the formula:\n$$\npq = \frac{(-7 + 24i) - (10 - 24i)}{2}\n$$\nSimplify the numerator:\n$$\n-7 + 24i - 10 + 24i = -17 + 48i\n$$\nNow divide by 2:\n$$\npq = \frac{-17 + 48i}{2} = -\frac{17}{2} + 24i\n$$", "## Conclusion: Final Answer", "Thus, the product $ pq $ is:\n$$\n\boxed{-\frac{17}{2} + 24i}\n$$", "This method efficiently leverages algebraic identities to solve for unknowns in complex number systems—no need for full cubic factorization. Understanding such techniques strengthens problem-solving skills in complex analysis, signal processing, and quantum mechanics, where complex arithmetic is fundamental.", "---", "Key SEO Elements:\n- Clear title optimized for search intent\n- Step-by-step explanation with mathematical reasoning\n- Natural inclusion of primary keywords: “complex numbers,” “find $ pq $,” “identity,” “$ p + q $, $ p^2 + q^2 $”\n- Focused on utility and conceptual insight for readers\n- Clean, readable formatting with logical progression", "This article helps both learners master a core identity and educators present it clearly, suitable for technical blogs, math forums, or online courses."]
