Let the integers be x and x+1. Then, (x+1)² - x² = 35.
["Title: Solve the Simple Algebraic Equation: Let Integers Be x and x+1|| | (x+1)² – x² = 35", "Meta Description:\nDiscover how letting two consecutive integers be represented as ( x ) and ( x+1 ) simplifies the equation ( (x+1)^2 - x^2 = 35 ). Learn step-by-step how to solve this classic algebra problem and uncover its real-world significance.", "---", "## Let the Integers Be ( x ) and ( x+1 ), Then Solve ( (x+1)² - x² = 35 )", "Mathematics often begins with simple, powerful ideas. One such idea is representing consecutive integers as ( x ) and ( x+1 ). This simple substitution opens the door to solving elegant algebraic expressions — like the one below:", "[\n(x+1)^2 - x^2 = 35\n]", "This equation appears simple, but it reveals key algebraic principles and is a classic example of how patterns in numbers solve quadratic expressions. Let’s break this down.", "---", "### Step 1: Expand the Expression", "Start by expanding ( (x+1)^2 ):", "[\n(x+1)^2 = x^2 + 2x + 1\n]", "Now substitute back into the original equation:", "[\nx^2 + 2x + 1 - x^2 = 35\n]", "---", "### Step 2: Simplify the Equation", "Notice that ( x^2 ) cancels out:", "[\n2x + 1 = 35\n]", "---", "### Step 3: Solve for ( x )", "Subtract 1 from both sides:", "[\n2x = 34\n]", "Divide by 2:", "[\nx = 17\n]", "---", "### Step 4: Find the Consecutive Integers", "Since we defined the integers as ( x = 17 ) and ( x+1 = 18 ), the pair ( (17, 18) ) satisfies the original equation.", "---", "### Why This Equation Matters: A Quick Breakdown", "This equation is a perfectly valid way to explore quadratic differences between consecutive integers. Expanding gives:", "[\n(x+1)^2 - x^2 = x^2 + 2x + 1 - x^2 = 2x + 1\n]", "So,\n[\n2x + 1 = 35 \quad \Rightarrow \quad x = 17\n]", "This proves that the difference between the squares of any two consecutive integers ( x ) and ( x+1 ) is always ( 2x + 1 ). Setting this equal to 35 is not only solvable but illustrates a core algebraic identity.", "---", "### Real-World and Theoretical Applications", "- Algebraic Foundations: This problem teaches how expanding and simplifying expressions leads to solutions.\n- Number Theory: It shows how consecutive integers form a simple yet profound arithmetic relationship.\n- Problem Solving Strategy: Using substitution to simplify real-world variables into abstract numbers is a widely applicable technique.\n- Educational Tool: This classic example is often used in classrooms to introduce quadratic reasoning and algebraic manipulation.", "---", "In summary:\nBy letting the integers be ( x ) and ( x+1 ), the equation ( (x+1)^2 - x^2 = 35 ) becomes a clear, solvable linear equation. Solving gives ( x = 17 ), and the consecutive integers are ( 17 ) and ( 18 ). This elegant approach highlights the beauty and utility of algebra in connecting numbers through patterns.", "---", "Want to master more such algebraic puzzles? Explore related topics like Diophantine equations, quadratic identities, and number series to deepen your mathematical insight.", "---", "Keywords:\nlet integers be x and x+1, solve (x+1)² – x² = 35, algebra problem, consecutive integers solution, expand (x+1)², apply algebra, quadratic difference, solve 2x + 1 = 35, integer difference identity", "Ready to solve more equations? Check out similar step-by-step algebraic strategies for instant results!"]
