Solution: Use identity $ p^2 + q^2 = (p + q)^2 - 2pq $. Substituting: $ 10 - 24i = (3 + 4i)^2 - 2pq $. Compute $ (3 + 4i)^2 = 9 + 24i - 16 = -7 + 24i $. Then $ 10 - 24i = -7 + 24i - 2pq $. Rearranging: $ 17 - 48i = -2pq \Rightarrow pq = \frac{-17 + 48i}{2} $. \boxed{-\dfrac{17}{2} + 24i}
["Mastering Complex Numbers: Using the Identity $ p^2 + q^2 = (p + q)^2 - 2pq $", "When solving equations and manipulating complex numbers, one powerful identity is:", "$$\np^2 + q^2 = (p + q)^2 - 2pq\n$$", "This formula allows us to transform sums of squares into simpler expressions involving sums and products—especially useful when working with imaginary components. Let’s explore how this identity helps solve complex number equations with clarity and precision.", "---", "### The Identity Explained", "Start with the foundational identity:", "$$\np^2 + q^2 = (p + q)^2 - 2pq\n$$", "This means that instead of computing $ p^2 + q^2 $ directly, we can use known values of $ p + q $ and $ (p + q)^2 $, then subtract twice the product $ 2pq $ to recover the expression.", "---", "### Example Application with Complex Numbers", "Let’s apply this method to a complex equation:", "$$\n10 - 24i = (3 + 4i)^2 - 2pq\n$$", "First, compute $ (3 + 4i)^2 $:", "$$\n(3 + 4i)^2 = 3^2 + 2 \cdot 3 \cdot 4i + (4i)^2 = 9 + 24i - 16 = -7 + 24i\n$$", "Now substitute back:", "$$\n10 - 24i = (-7 + 24i) - 2pq\n$$", "Rearranging the equation:", "$$\n10 - 24i + 7 - 24i = -2pq \Rightarrow 17 - 48i = -2pq\n$$", "Now solve for $ pq $:", "$$\npq = \frac{17 - 48i}{-2} = -\frac{17}{2} + 24i\n$$", "---", "### Final Result: $ pq = -\dfrac{17}{2} + 24i $", "This elegant method reduces complex calculations into manageable steps by leveraging identities. Recognizing patterns like $ p^2 + q^2 $ can simplify problem-solving in algebra, number theory, and even engineering applications involving complex systems.", "---", "Why This Matters:", "- Efficiently handles complex arithmetic by breaking down expressions\n- Saves time and reduces errors in computational math\n- Builds deeper understanding of algebraic structures with imaginary numbers", "Whether you're a student tackling complex equations or a coder working with vector-based complex numbers, mastering this identity accelerates your ability to reason and compute confidently.", "---", "Try It Yourself:\nPick another complex expression and use $ p^2 + q^2 = (p + q)^2 - 2pq $ to solve for unknowns—it’s faster than brute force!", "---", "Summary:\nBy applying the identity $ p^2 + q^2 = (p + q)^2 - 2pq $ and strategic substitution, we simplified the complex equation and found:", "$$\n\boxed{-\dfrac{17}{2} + 24i}\n$$", "A clear, scalable technique for complex number manipulation—essential for math mastery!"]
