The sum of the squares of two consecutive positive integers is 145. What is the larger integer?
["The Sum of the Squares of Two Consecutive Positive Integers Is 145: What Is the Larger Integer?", "Finding simple yet meaningful solutions to math problems can deepen your understanding of algebra and number relationships. One classic problem that often puzzles curious minds is:\nWhat are the two consecutive positive integers whose squares add up to 145?", "Let’s solve this step-by-step while learning key math concepts along the way.", "---", "### Step 1: Define the Variables\nLet the first consecutive positive integer be ( x ).\nThen, the next consecutive positive integer is ( x + 1 ).", "---", "### Step 2: Formulate the Equation\nThe problem states:\n[\nx^2 + (x + 1)^2 = 145\n]", "Expand the right-hand side:\n[\nx^2 + (x^2 + 2x + 1) = 145\n]", "Combine like terms:\n[\n2x^2 + 2x + 1 = 145\n]", "---", "### Step 3: Simplify the Equation\nSubtract 145 from both sides:\n[\n2x^2 + 2x + 1 - 145 = 0\n]\n[\n2x^2 + 2x - 144 = 0\n]", "Divide the entire equation by 2 to simplify:\n[\nx^2 + x - 72 = 0\n]", "---", "### Step 4: Solve the Quadratic Equation\nFactor the quadratic:\nWe seek two numbers that multiply to (-72) and add to (1). Those numbers are (9) and (-8):\n[\n(x + 9)(x - 8) = 0\n]", "Set each factor equal to zero:\n[\nx + 9 = 0 \quad \Rightarrow \quad x = -9 \quad (\ ext{discard, since only positive integers are considered})\n]\n[\nx - 8 = 0 \quad \Rightarrow \quad x = 8\n]", "---", "### Step 5: Identify the Integers and Final Answer\nIf ( x = 8 ), then the two consecutive positive integers are:\n[\n8 \quad \ ext{and} \quad 9\n]", "Check:\n[\n8^2 + 9^2 = 64 + 81 = 145 \quad \ ext{(Correct!)}\n]", "Thus, the larger integer is:\n[\n\boxed{9}\n]", "---", "### Bonus Insight: Why This Method Works\nThis problem demonstrates how forming a quadratic equation from real-world relationships allows precise solutions. Recognizing that consecutive integers differ by 1 streamlines simplification and factoring — essential skills in algebra.", "---", "Summary:\n- Let the integers be ( x ) and ( x + 1 ).\n- Set up: ( x^2 + (x+1)^2 = 145 ).\n- Solve: ( x = 8 ), so the larger integer is 9.", "Remember, math problems like this are not just about answers — they’re about logical reasoning and structured thinking. Keep practicing, and you’ll master these elegant relationships in no time!"]
