An archaeologist is examining an ancient Mesopotamian tablet that records two quantities related to grain storage: their sum is 15, and their product is 50. Determine the sum of the squares of these two quantities.

["Title: Unlocking Ancient Trade: Solving for Grains in a Mesopotamian Tablet\nKeywords: Mesopotamian tablet, ancient grain storage, sum and product, algebra, archaeologist findings", "---", "In a remarkable discovery, an archaeologist examining a centuries-old Mesopotamian clay tablet has uncovered a mathematical puzzle inscribed over 3,700 years ago. The tablet presents a classic problem involving two quantities tied to grain storage—specifically, their sum and product—offering a striking glimpse into ancient administrative practices and mathematical reasoning.", "According to the tablet, the sum of two unknown quantities is 15, and their product is 50. While the precise quantities are not explicitly labeled, historians and mathematicians can reconstruct them using algebraic methods rooted in ancient Mesopotamian arithmetic.", "Let the two quantities be ( x ) and ( y ). The clues from the tablet are:\n- ( x + y = 15 )\n- ( xy = 50 )", "The question of interest is: What is the sum of the squares of these quantities?\nThat is, what is ( x^2 + y^2 )?", "Modern algebraic identity provides a powerful shortcut:\n[\nx^2 + y^2 = (x + y)^2 - 2xy\n]", "Substituting the known values:\n- ( x + y = 15 \Rightarrow (x + y)^2 = 15^2 = 225 )\n- ( xy = 50 \Rightarrow 2xy = 2 \ imes 50 = 100 )", "So:\n[\nx^2 + y^2 = 225 - 100 = 125\n]", "Thus, the sum of the squares of the two grain quantities inscribed on the ancient tablet is 125. This calculation not only deciphers a historical mystery but also highlights the sophistication of early mathematics in Mesopotamia—where traders and administrators used precise numerical reasoning long before modern algebra.", "This tablet stands as a testament to humanity’s enduring desire to measure, record, and understand the world—one grain at a time.", "---", "Significance for archaeologists and historians:\nBeyond its numerical resolution, the tablet reflects the organizational complexity of ancient Mesopotamian economies. Records like these were critical for managing food supplies, tax collection, and trade—cornerstones of early urban civilizations.", "Related searches:\n- Ancient Mesopotamian mathematics\n- Clay tablet archaeology\n- How ancient civilizations calculated grain storage\n- Sum and product from ancient tablets", "---", "Note: This article uses inferred numerical values from a reconstructed mathematical cipher; actual ancient tablets may vary, but the algebraic approach remains consistent across Mesopotamian record-keeping."]