If \( x \) and \( y \) are positive numbers such that \( x + y = 10 \) and \( xy = 21 \), what is the value of \( x^2 + y^2 \)?

["Title: How to Find ( x^2 + y^2 ) When ( x + y = 10 ) and ( xy = 21 ) – A Step-by-Step Explanation", "When solving problems involving two variables constrained by both a sum and a product, a powerful algebraic identity helps simplify calculations:", "[\nx^2 + y^2 = (x + y)^2 - 2xy\n]", "This formula is especially useful when you're given ( x + y ) and ( xy ), as in many real-world and mathematical scenarios. Let’s apply it to the given conditions in this problem.", "We are told:\n- ( x + y = 10 )\n- ( xy = 21 )", "Now plug these values into the identity:", "[\nx^2 + y^2 = (x + y)^2 - 2xy = (10)^2 - 2(21)\n]", "Calculate each term:\n- ( (10)^2 = 100 )\n- ( 2 \ imes 21 = 42 )", "Subtract:\n[\n100 - 42 = 58\n]", "So, the value of ( x^2 + y^2 ) is ( 58 ).", "### Why This Method Works\nThis approach avoids the need to solve quadratic equations or find ( x ) and ( y ) individually, making it faster and more efficient. By using clever algebraic identities, even complex relations between variables collapse into simple arithmetic.", "### Real-World Application\nSuch problems appear in physics, economics, and optimization. For example, determining total energy or power from combined and product-based constraints allows engineers and data scientists to simplify complex models.", "### Summary\nGiven ( x + y = 10 ) and ( xy = 21 ),\n[\nx^2 + y^2 = (x + y)^2 - 2xy = 10^2 - 2 \cdot 21 = 100 - 42 = 58\n]", "Answer: ( x^2 + y^2 = 58 )", "---", "Keywords: ( x^2 + y^2 ), ( x + y = 10 ), ( xy = 21 ), algebra, identity, solving equations, positive numbers, mathematical identity, real-world applications.\nMeta description: Learn how to efficiently calculate ( x^2 + y^2 ) using the identity ( x^2 + y^2 = (x + y)^2 - 2xy ) when ( x + y = 10 ) and ( xy = 21 ). Step-by-step explanation with examples and applications."]