\( 4n^2 + 4n^2 + 8n + 4 = 340 \)

["# Solving the Quadratic Equation: ( 4n^2 + 4n^2 + 8n + 4 = 340 )", "Solving quadratic equations is a fundamental skill in algebra, vital for students, educators, and math enthusiasts. One such equation appearing in academic contexts is:", "[\n4n^2 + 4n^2 + 8n + 4 = 340\n]", "At first glance, this appears simplified to ( 8n^2 + 8n + 4 = 340 ), but mastering its full resolution helps sharpen algebraic techniques and problem-solving strategies. This article explains how to simplify, rearrange, and solve this quadratic equation step-by-step.", "---", "## Step 1: Simplify the Left-Hand Side", "The given equation is:", "[\n4n^2 + 4n^2 + 8n + 4 = 340\n]", "Combine like terms:", "[\n(4n^2 + 4n^2) + 8n + 4 = 340\n]", "[\n8n^2 + 8n + 4 = 340\n]", "This confirms the simplified form:\n( 8n^2 + 8n + 4 = 340 )", "---", "## Step 2: Move All Terms to One Side to Form a Standard Quadratic", "Subtract 340 from both sides:", "[\n8n^2 + 8n + 4 - 340 = 0\n]", "[\n8n^2 + 8n - 336 = 0\n]", "---", "## Step 3: Simplify the Equation by Dividing by the Greatest Common Factor", "The coefficients (8), (8), and (-336) share a common divisor: 8.", "Divide the entire equation by 8:", "[\nn^2 + n - 42 = 0\n]", "Now we have a simplified standard quadratic equation:\n( n^2 + n - 42 = 0 )", "---", "## Step 4: Factor the Quadratic (If Possible)", "Look for two integers whose product is (-42) and sum is (+1).", "Factors of (-42):\n- (7 \ imes (-6) = -42), and (7 + (-6) = 1) → perfect!", "Thus, the equation factors as:", "[\n(n + 7)(n - 6) = 0\n]", "---", "## Step 5: Solve for ( n )", "Set each factor equal to zero:", "[\nn + 7 = 0 \quad \Rightarrow \quad n = -7\n]", "[\nn - 6 = 0 \quad \Rightarrow \quad n = 6\n]", "---", "## Step 6: Interpret the Solutions", "The equation ( 4n^2 + 4n^2 + 8n + 4 = 340 ) has two real solutions:\n[\nn = -7 \quad \ ext{and} \quad n = 6\n]", "In practical contexts, such as modeling motion or profit calculations, negative solutions may have limited relevance depending on the scenario’s domain. However, algebraically both are valid.", "---", "## Step 7: Verification", "Plug ( n = 6 ) back into the original equation:", "[\n4(6)^2 + 4(6)^2 + 8(6) + 4 = 4(36) + 4(36) + 48 + 4 = 144 + 144 + 48 + 4 = 340 \quad ✔\n]", "Plug ( n = -7 ):", "[\n4(-7)^2 + 4(-7)^2 + 8(-7) + 4 = 4(49) + 4(49) - 56 + 4 = 196 + 196 - 56 + 4 = 340 \quad ✔\n]", "Both values satisfy the equation.", "---", "## Bonus: Analytical Solution Using the Quadratic Formula", "For completeness, apply the quadratic formula:", "[\nn = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n]", "With ( a = 1 ), ( b = 1 ), ( 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\sqrt{169} = 13 \Rightarrow n = \frac{-1 \pm 13}{2}\n]", "[\nn = \frac{12}{2} = 6 \quad \ ext{or} \quad n = \frac{-14}{2} = -7\n]", "Matches earlier factorization.", "---", "## Conclusion", "The equation ( 4n^2 + 4n^2 + 8n + 4 = 340 ) simplifies elegantly into a solvable quadratic form. Through combining like terms, rearranging, simplifying coefficients, and applying factoring and the quadratic formula, we confidently find two solutions: ( n = -7 ) and ( n = 6 ).", "Mastering such algebraic manipulations strengthens analytical reasoning and prepares learners for advanced topics. Whether used in math class, competitive exams, or real-world modeling, quadratic equations like this illustrate the power of structured problem-solving.", "---", "### Keywords:\n- Solve ( 4n^2 + 4n^2 + 8n + 4 = 340 )\n- Quadratic equation solving\n- Algebraic simplification\n- ( n^2 + n - 42 = 0 )\n- Factoring quadratic equations\n- Quadratic formula application\n- Step-by-step quadratic solution", "---", "Tip: Practice transforming expressions like ( 4n^2 + 4n^2 ) — recognizing like terms is essential for efficient solving."]