If the sum of the squares of two consecutive integers is 85, what are the integers?
["Title: Solve This Simple Math Puzzle: If the Sum of Squares of Two Consecutive Integers Is 85, What Are They?", "If you’re curious about simple yet engaging math problems, one classic puzzle stirs interest: What are two consecutive integers whose squares add up to 85? This question not only sharpens problem-solving skills but also illustrates how algebra helps uncover patterns in numbers.", "In this article, we’ll explore how to set up the equation, solve for the integers, and uncover the interesting properties of consecutive numbers.", "---", "### Understanding the Problem", "Two consecutive integers mean two whole numbers that follow each other without gaps—like 8 and 9, or -4 and -3. Let’s define them algebraically.", "Let the smaller integer be:\nx", "Then the next consecutive integer is:\nx + 1", "We are told that the sum of their squares equals 85. So, the equation becomes:", "[\nx^2 + (x + 1)^2 = 85\n]", "---", "### Step-by-Step Solution", "1. Expand the equation:\n[\nx^2 + (x^2 + 2x + 1) = 85\n]", "2. Combine like terms:\n[\n2x^2 + 2x + 1 = 85\n]", "3. Move all terms to one side:\n[\n2x^2 + 2x + 1 - 85 = 0 \implies 2x^2 + 2x - 84 = 0\n]", "4. Simplify the equation:\nDivide every term by 2:\n[\nx^2 + x - 42 = 0\n]", "5. Solve the quadratic using factoring:\nWe look for two numbers that multiply to –42 and add to +1. Those numbers are +7 and –6:\n[\n(x + 7)(x - 6) = 0\n]", "6. Apply the zero-factor principle:\nSet each factor to zero:\n[\nx + 7 = 0 \implies x = -7\n]\n[\nx - 6 = 0 \implies x = 6\n]", "So the possible values for the smaller integer are –7 and 6.", "---", "### Finding the Integers", "- If ( x = 6 ), the integers are 6 and 7\n Check: ( 6^2 + 7^2 = 36 + 49 = 85 ) ✅\n- If ( x = -7 ), the integers are –7 and –6\n Check: ( (-7)^2 + (-6)^2 = 49 + 36 = 85 ) ✅", "---", "### Why This Problem Matters", "This problem highlights:", "- Algebraic modeling: Translating word problems into equations.\n- Integer properties: The uniqueness and symmetry of consecutive integers.\n- Multiple solutions: Sometimes, a single condition allows for two valid solutions (positive and negative).", "It’s a fantastic exercise for students learning algebra or anyone fascinated by number patterns.", "---", "### Final Answer", "The two consecutive integers whose squares sum to 85 are 6 and 7, or –7 and –6.", "---", "Related Keywords for SEO:\nconsecutive integers sum of squares, solve for consecutive integers equation, math puzzle integers, algebra word problems, consecutive integers 85, how to find two consecutive integers with square sum 85, integer roots equation, math problem explanation, consecutive numbers and algebra.", "---", "Download & Master the Technique:\nTry solving similar problems using algebra to build confidence in translating math stories into equations. Whether you're learning for school or just love numbers, this classic puzzle reveals how math connects logic and pattern recognition."]
