The sum of the squares of two consecutive integers is 365. What are the integers?
["The Sum of the Squares of Two Consecutive Integers Is 365\nSolving the Equation: What Are the Integers?", "Finding two consecutive integers whose squares add up to 365 is a classic math puzzle that combines algebra and number theory. Whether you’re a student tackling homework or a math enthusiast exploring number patterns, this problem offers an engaging challenge. In this article, we’ll dive into how to solve it step by step and reveal the two integers that satisfy this intriguing condition.", "---", "### What Are Consecutive Integers?", "Consecutive integers are numbers that follow each other in sequence without gaps. For example, if one integer is n, the next is n + 1. The problem states that the sum of the squares of two consecutive integers equals 365:", "[\nn^2 + (n + 1)^2 = 365\n]", "---", "### Setting Up the Equation", "Start by expanding and simplifying the expression:", "[\nn^2 + (n + 1)^2 = 365\n]", "[\nn^2 + (n^2 + 2n + 1) = 365\n]", "Combine like terms:", "[\n2n^2 + 2n + 1 = 365\n]", "Subtract 365 from both sides:", "[\n2n^2 + 2n + 1 - 365 = 0\n]", "[\n2n^2 + 2n - 364 = 0\n]", "---", "### Simplify the Quadratic Equation", "Divide the entire equation by 2 to reduce complexity:", "[\nn^2 + n - 182 = 0\n]", "Now solve this quadratic equation using the quadratic formula:", "[\nn = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "Here, (a = 1), (b = 1), and (c = -182):", "[\nn = \frac{-1 \pm \sqrt{1^2 - 4(1)(-182)}}{2(1)}\n]", "[\nn = \frac{-1 \pm \sqrt{1 + 728}}{2}\n]", "[\nn = \frac{-1 \pm \sqrt{729}}{2}\n]", "[\nn = \frac{-1 \pm 27}{2}\n]", "---", "### Find the Two Possible Solutions", "Calculate both roots:", "1. ( n = \frac{-1 + 27}{2} = \frac{26}{2} = 13 )\n2. ( n = \frac{-1 - 27}{2} = \frac{-28}{2} = -14 )", "Both values are integers, so both pairs are valid.", "---", "### Identify the Consecutive Integers", "- If ( n = 13 ), the integers are 13 and 14\n- If ( n = -14 ), the integers are –14 and –13", "Let’s verify both to confirm they satisfy the original condition:", "- (13^2 + 14^2 = 169 + 196 = 365)\n- ((-14)^2 + (-13)^2 = 196 + 169 = 365)", "Both pairs work.", "---", "### Why This Problem Matters", "This problem illustrates how algebra helps uncover hidden patterns in integers. It’s often used in math education to teach:", "- Setting up equations from word problems\n- Expanding and simplifying expressions\n- Solving quadratic equations logically", "Moreover, it underscores that integer solutions can come in both positive and negative forms—showing math’s universality.", "---", "### Final Answer", "The two consecutive integers whose squares sum to 365 are:", "[\n\boxed{13 \ ext{ and } 14 \quad \ ext{or} \quad -14 \ ext{ and } -13}\n]", "Whether you’re exploring such puzzles for fun or as a math learner, remember: behind every sum of squares lies a simple but elegant equation waiting to be solved.", "---", "Keywords: sum of squares of two consecutive integers, integer solution, algebra problem, 365 squared numbers, quadratic equation integers, math problem, consecutive integers equation", "Meta Description:\nDiscover how to solve the equation representing two consecutive integers whose squares sum to 365. Learn step-by-step how (n^2 + (n+1)^2 = 365) yields two valid integer pairs: 13 and 14, or –14 and –13. Perfect for math students and enthusiasts."]
