["Solving the Equation n² + (n + 1)² = 85: A Step-by-Step Guide with Practical Insights", "If you’ve ever come across the quadratic expression n² + (n + 1)² = 85, you’re not alone — this equation is a popular puzzle in algebra, math competitions, and standardized test preparation. In this article, we’ll explore how to solve it step-by-step, understand its solutions, and discuss why it matters in math learning and problem-solving.", "---", "### Understanding the Equation: n² + (n + 1)² = 85", "The equation combines two perfect squares:
\n- n²: the square of an integer (or real number) n,
\n- (n + 1)²: the square of the next consecutive integer.", "This expression arises frequently when exploring patterns in consecutive numbers, number sequences, and algebraic manipulations.", "---", "### Step 1: Expand and Simplify", "Let’s begin by expanding the equation:", "[
\nn^2 + (n + 1)^2 = n^2 + (n^2 + 2n + 1) = 2n^2 + 2n + 1
\n]", "So the equation becomes:", "[
\n2n^2 + 2n + 1 = 85
\n]", "---", "### Step 2: Bring All Terms to One Side", "Subtract 85 from both sides:", "[
\n2n^2 + 2n + 1 - 85 = 0
\n]", "[
\n2n^2 + 2n - 84 = 0
\n]", "---", "### Step 3: Simplify the Quadratic Equation", "Divide every term by 2 to reduce complexity:", "[
\nn^2 + n - 42 = 0
\n]", "This is now a standard quadratic equation in the form an² + bn + c = 0, with:
\n- a = 1,
\n- b = 1,
\n- c = -42", "---", "### Step 4: Solve Using the Quadratic Formula", "The quadratic formula is:", "[
\nn = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\n]", "Plugging in values:", "[
\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 √169 = 13, we get:", "[
\nn = \frac{-1 \pm 13}{2}
\n]", "This gives two potential solutions:", "- ( n = \frac{-1 + 13}{2} = \frac{12}{2} = 6 )
\n- ( n = \frac{-1 - 13}{2} = \frac{-14}{2} = -7 )", "---", "### Step 5: Verify Both Solutions", "It’s essential to plug values back into the original equation to confirm validity.", "For n = 6:", "[
\n6^2 + (6 + 1)^2 = 36 + 49 = 85 \quad \ ext{(Valid)}
\n]", "For n = -7:", "[
\n(-7)^2 + (-7 + 1)^2 = 49 + (-6)^2 = 49 + 36 = 85 \quad \ ext{(Also valid)}
\n]", "Both integers — 6 and -7 — satisfy the equation.", "---", "### Why This Problem Matters", "- Pattern Recognition: It helps identify patterns in consecutive integers and their squares.
\n- Foundation for Higher Math: Understanding such equations prepares students for polynomial solving, irrational numbers, and algebra-based problem solving.
\n- Common in Puzzles & Tests: This equation frequently appears in math Olympiads, SAT problems, and logic puzzles.", "---", "### Conclusion", "Solving n² + (n + 1)² = 85 is a simple yet powerful exercise in algebra that strengthens fundamental math skills. Whether you're tackling it for homework, a math competition, or curiosity, recognizing both solutions — n = 6 and n = -7 — affirms the beauty and symmetry in mathematical equations.", "Next time you encounter a similar equation, follow the same clear steps: expand, simplify, solve using the quadratic formula, and verify — and watch the logic unfold!", "---", "Keywords: n² + (n+1)² = 85, algebra solution, quadratic equation, solve n² + (n+1)² = 85, consecutive integers equation, math problem solving, quadratic formula, algebra tutorial", "Meta Description: Solve n² + (n + 1)² = 85 step by step. Learn algebra techniques, verify both integer solutions (n = 6 and n = -7), and explore applications in math learning."]
