Equation: x² + (x+1)² = 145 → x² + x² + 2x + 1 = 145 → 2x² + 2x + 1 = 145

["Solving the Quadratic Equation: x² + (x+1)² = 145 Step-by-Step", "Mathematics often reveals elegant solutions hidden within simple expressions. One common challenge students face is equations involving perfect squares, such as:", "[\nx^2 + (x+1)^2 = 145\n]", "In this article, we’ll walk through solving this quadratic equation step-by-step, uncovering the value of ( x ) and understanding the broader mathematical principles involved.", "---", "### How to Solve ( x^2 + (x+1)^2 = 145 )", "Start with the original equation:", "[\nx^2 + (x+1)^2 = 145\n]", "### Step 1: Expand the squared term", "First, expand ( (x+1)^2 ):", "[\n(x+1)^2 = x^2 + 2x + 1\n]", "Substitute into the equation:", "[\nx^2 + (x^2 + 2x + 1) = 145\n]", "### Step 2: Combine like terms", "Combine the ( x^2 ) and ( x^2 ) terms:", "[\n2x^2 + 2x + 1 = 145\n]", "### Step 3: Move all terms to one side", "Subtract 145 from both sides to form a standard quadratic equation:", "[\n2x^2 + 2x + 1 - 145 = 0\n]\n[\n2x^2 + 2x - 144 = 0\n]", "### Step 4: Simplify the equation", "Divide the entire equation by 2 to reduce coefficients:", "[\nx^2 + x - 72 = 0\n]", "---", "### Step 5: Solve the quadratic equation", "Use the quadratic formula:", "[\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "Here, ( a = 1 ), ( b = 1 ), and ( c = -72 ). Plug in the values:", "[\nx = \frac{-1 \pm \sqrt{1^2 - 4(1)(-72)}}{2(1)}\n]\n[\nx = \frac{-1 \pm \sqrt{1 + 288}}{2}\n]\n[\nx = \frac{-1 \pm \sqrt{289}}{2}\n]\n[\nx = \frac{-1 \pm 17}{2}\n]", "This gives two possible solutions:", "[\nx = \frac{-1 + 17}{2} = \frac{16}{2} = 8\n]\n[\nx = \frac{-1 - 17}{2} = \frac{-18}{2} = -9\n]", "---", "### Final Answer:", "The solutions are:", "[\nx = 8 \quad \ ext{or} \quad x = -9\n]", "---", "### Why This Equation Matters", "The equation ( x^2 + (x+1)^2 = 145 ) is more than just a quadratic puzzle — it reflects patterns in integer solutions, distance-like expressions (representing sum of squares), and teaches algebraic transformation and simplification skills crucial for higher mathematics.", "---", "### Summary", "- Expand and simplify to standard form\n- Apply the quadratic formula when factoring is difficult\n- Verify both solutions by substitution\n- Understand how algebraic identities help solve real-world and abstract problems", "If you’re tackling similar equations, practicing expansion, combining terms, and quadratic solving techniques will make the process faster and more intuitive.", "---", "Keywords: quadratic equation, solve ( x^2 + (x+1)^2 = 145 ), step-by-step solution, algebraic simplification, solve quadratic, sum of squares equation, mathematical techniques in algebra.", "---", "Explore more algebraic methods and equation-solving strategies to strengthen your math foundation and tackle complex problems with confidence!"]