\( (2n)^2 + (2n + 2)^2 = 340 \)

["# Solving the Equation: ( (2n)^2 + (2n + 2)^2 = 340 )", "Afternoon readers! If you're seeking a clear and concise route to solve quadratic expressions like ( (2n)^2 + (2n + 2)^2 = 340 ), you’re in the right place. This article walks you through solving this equation step-by-step, exploring important math concepts, and uncovering practical insights—all optimized for search engines.", "## Understanding the Equation", "The equation:\n[\n(2n)^2 + (2n + 2)^2 = 340\n]", "represents the sum of the squares of two linear expressions in ( n ):\n- ( (2n)^2 ),\n- and ( (2n + 2)^2 ).", "Expanding and simplifying such expressions allows us to transform word problems into solvable algebraic equations—perfect for students, math enthusiasts, or anyone interested in algebraic problem-solving.", "---", "## Step-by-Step Solution", "### Step 1: Expand the Squares", "First, expand each squared term:\n[\n(2n)^2 = 4n^2\n]\n[\n(2n + 2)^2 = (2n)^2 + 2 \cdot 2n \cdot 2 + 2^2 = 4n^2 + 8n + 4\n]", "### Step 2: Combine and Simplify", "Add the two expanded terms:\n[\n4n^2 + (4n^2 + 8n + 4) = 340\n]\n[\n8n^2 + 8n + 4 = 340\n]", "### Step 3: Bring Equation to Standard Quadratic Form", "Subtract 340 from both sides:\n[\n8n^2 + 8n + 4 - 340 = 0\n]\n[\n8n^2 + 8n - 336 = 0\n]", "### Step 4: Simplify the Quadratic Equation", "Divide the entire equation by 8 to reduce coefficients:\n[\nn^2 + n - 42 = 0\n]", "---", "## Step 5: Solve the Quadratic Equation", "Use the quadratic formula ( n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ) with ( a = 1 ), ( b = 1 ), and ( c = -42 ):", "[\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]\n[\n\sqrt{169} = 13 \Rightarrow n = \frac{-1 \pm 13}{2}\n]", "So, two solutions arise:", "[\nn = \frac{-1 + 13}{2} = \frac{12}{2} = 6\n]\n[\nn = \frac{-1 - 13}{2} = \frac{-14}{2} = -7\n]", "---", "## Final Thoughts and Validation", "Plugging ( n = 6 ) back into the original equation:", "[\n(2 \cdot 6)^2 + (2 \cdot 6 + 2)^2 = 12^2 + 14^2 = 144 + 196 = 340 \quad \ ext{(✓)}\n]", "Plugging ( n = -7 ):", "[\n(2 \cdot -7)^2 + (2 \cdot -7 + 2)^2 = (-14)^2 + (-12)^2 = 196 + 144 = 340 \quad \ ext{(✓)}\n]", "Both values are valid solutions.", "---", "## Key Takeaways", "- Expressions like ( (2n)^2 + (2n+2)^2 ) often appear in word problems involving patterns or optimization.\n- Expanding and simplifying before solving transforms complexity into standard quadratic form.\n- The quadratic formula provides a reliable method even when factoring is difficult.\n- Always verify solutions by substituting back—this ensures accuracy and deepens understanding.", "---", "## Why This Equation Matters", "Equations like ( (2n)^2 + (2n + 2)^2 = 340 ) model real-life situations—from optimizing area and perimeter combinations to analyzing sequences and series. Mastering such problems builds critical thinking and algebraic fluency essential for STEM fields and advanced math.", "---", "## Ready to Practice?", "Try solving your own variations:", "- Change the constant to ( 300 ) or ( 400 ) and see how solutions shift.\n- Replace ( 2n ) with a different coefficient (e.g., ( 3n )) to explore scaling effects.\n- Combine with other perfect squares to create multi-term equations.", "---", "### Related SEO Keywords:\nsolve (2n)^2 + (2n+2)^2 = 340, quadratic equation solution, algebra problem step-by-step, solving equations with perfect squares, common core math problems", "---", "Keywords optimized for students, educators, and learners searching for clear algebraic problem-solving guides.", "---", "Keywords: ((2n)^2 + (2n + 2)^2 = 340), quadratic equation, algebra problem solving, solved quadratic, math tutorial, equation simplification, common core math, step-by-step math explanation."]