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

["Title: Solve for ( x^2 + y^2 ) Given ( x + y = 10 ) and ( xy = 21 )", "Understanding algebraic identities can simplify complex problems — and this is a perfect example of using key formulas to solve for expressions like ( x^2 + y^2 ) efficiently.", "### Understanding the Problem", "We are given two equations involving two numbers ( x ) and ( y ):\n1. ( x + y = 10 )\n2. ( xy = 21 )", "We are asked to find the value of ( x^2 + y^2 ). While directly computing squares might seem tricky, an important identity in algebra provides a shortcut.", "### Use the Algebraic Identity", "Recall the identity:\n[\nx^2 + y^2 = (x + y)^2 - 2xy\n]\nThis identity expresses ( x^2 + y^2 ) in terms of the known sum and product.", "### Substitute Known Values", "From the problem:\n- ( x + y = 10 \Rightarrow (x + y)^2 = 10^2 = 100 )\n- ( xy = 21 \Rightarrow 2xy = 2 \ imes 21 = 42 )", "Now substitute into the identity:\n[\nx^2 + y^2 = (x + y)^2 - 2xy = 100 - 42 = 58\n]", "### Final Answer", "Therefore, the value of ( x^2 + y^2 ) is:\n[\n\boxed{58}\n]", "### Why This Matters", "This simple yet powerful identity saves time and effort in solving quadratic-related problems. Whether you're working on algebra exams, coding math problems, or just want to grasp how relationships between numbers reveal deeper patterns — knowing ( x^2 + y^2 = (x + y)^2 - 2xy ) is essential.", "Remember: When dealing with sums and products, transforming expressions using algebraic identities like this is your key to quick and accurate solutions.", "---", "Keywords: ( x^2 + y^2 ) value, solve ( x + y = 10 ), solve ( xy = 21 ), algebraic identity, quadratic formula shortcut, math Tip, algebra shortcut, sum and product identity."]