The sum of the squares of two consecutive integers is 85. What are the integers?
["# The Sum of the Squares of Two Consecutive Integers Is 85 — What Are They?", "Mathematics often hides elegant solutions in plain sight, and one classic puzzle involves finding two consecutive integers whose squares add up to 85. Solving this dream team problem not only sharpens algebra skills but also reveals beautiful logic behind numbers. In this article, we’ll explore step-by-step how to find these integers and understand the general approach.", "## What Are Consecutive Integers?", "Consecutive integers are integers that follow one after the other without gaps—like 4 and 5, or -3 and -2. To work with them algebraically, we typically let the first integer be:", "$$ n $$", "Then the next consecutive integer is:", "$$ n + 1 $$", "## Setting Up the Equation", "We’re told the sum of the squares of these two integers equals 85:", "$$\nn^2 + (n + 1)^2 = 85\n$$", "### Expand and simplify", "First, expand $ (n + 1)^2 $:", "$$\nn^2 + (n^2 + 2n + 1) = 85\n$$", "Combine like terms:", "$$\n2n^2 + 2n + 1 = 85\n$$", "Subtract 85 from both sides:", "$$\n2n^2 + 2n + 1 - 85 = 0\n$$", "$$\n2n^2 + 2n - 84 = 0\n$$", "Divide the entire equation by 2 to simplify:", "$$\nn^2 + n - 42 = 0\n$$", "## Solve the Quadratic Equation", "Use the quadratic formula:", "$$\nn = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n$$", "Here, $ a = 1 $, $ b = 1 $, $ c = -42 $:", "$$\nn = \frac{-1 \pm \sqrt{1^2 - 4(1)(-42)}}{2(1)} = \frac{-1 \pm \sqrt{1 + 168}}{2} = \frac{-1 \pm \sqrt{169}}{2}\n$$", "Since $ \sqrt{169} = 13 $:", "$$\nn = \frac{-1 \pm 13}{2}\n$$", "This gives two solutions:", "- $ n = \frac{-1 + 13}{2} = \frac{12}{2} = 6 $\n- $ n = \frac{-1 - 13}{2} = \frac{-14}{2} = -7 $", "## Identify the Pairs of Consecutive Integers", "Using each solution:", "- If $ n = 6 $, the pair is $ 6 $ and $ 7 $:\n $ 6^2 + 7^2 = 36 + 49 = 85 $ ✅\n- If $ n = -7 $, the pair is $ -7 $ and $ -6 $:\n $ (-7)^2 + (-6)^2 = 49 + 36 = 85 $ ✅", "## Therefore, the two possible pairs of consecutive integers are:", "- $ 6 $ and $ 7 $\n- $ -7 $ and $ -6 $", "## Why Both Pairs Work", "The equation involves squaring, which removes sign differences. Since $ (-a)^2 = a^2 $, the sum of squares is the same whether the integers are positive or negative and consecutive.", "## Real-World and Logical Insight", "This problem illustrates a fundamental property: the sum of squares of consecutive integers remains valid across both positive and negative sequences. It also reflects symmetry and balance in number systems—how two adjacent numbers can perfectly combine to form a fixed total.", "## Conclusion", "The sum of the squares of two consecutive integers equals 85 when the integers are either:", "- $ \boxed{6} $ and $ \boxed{7} $, or\n- $ \boxed{-7} $ and $ \boxed{-6} $", "This problem is a simple yet powerful example of how algebra uncovers real number patterns, perfect for students, math enthusiasts, and anyone curious about number theory. Next time you explore equations, remember: sometimes the answer lies just between two adjacent numbers.", "---", "Keywords: sum of squares of two consecutive integers, solve $ n^2 + (n+1)^2 = 85 $, consecutive integers equation, integer solutions, algebra problem, math puzzle solution, quadratic equation step-by-step."]
