Subtract 85 from both sides: \( 2n^2 + 2n + 1 - 85 = 0 \), \( 2n^2 + 2n - 84 = 0 \).
["SEO-Optimized Article: Simplifying the Quadratic Equation: ( 2n^2 + 2n - 84 = 0 ) After Subtracting 85 from Both Sides", "---", "Understanding How to Simplify Quadratic Equations: A Step-by-Step Guide", "Working with quadratic equations is a fundamental skill in algebra, essential for students, educators, and math enthusiasts. One common task is simplifying equations by combining or eliminating constants, and a typical example involves transforming expressions like ( 2n^2 + 2n + 1 - 85 = 0 ) into a standard quadratic form. In this article, we explore how subtracting 85 from both sides leads to the simplified equation ( 2n^2 + 2n - 84 = 0 ), a key step in solving quadratic forms. We’ll also explain why this simplification matters and how it helps solve the equation efficiently.", "---", "### What Happens When We Subtract 85 from Both Sides?", "The original equation is:", "[\n2n^2 + 2n + 1 - 85 = 0\n]", "Subtracting 85 from both sides stabilizes the equation by eliminating constant terms on the left:", "[\n2n^2 + 2n + 1 - 85 - 85 = 0 - 85\n]", "Wait — mistakenly, the simplification is applied only to the constant term on the left-hand side, not adding extra subtraction. So correctly:", "[\n2n^2 + 2n + 1 - 85 = (2n^2 + 2n - 84)\n]", "Thus, the equation becomes:", "[\n2n^2 + 2n - 84 = 0\n]", "This transformation brings the equation into a standard quadratic form, making it easier to factor, apply the quadratic formula, or complete the square.", "---", "### Why Simplify Before Solving?", "Simplifying quadratic equations reduces complexity. Instead of handling terms scattered across the equation, combining constants like ( +1 - 85 ) to ( -84 ) clarifies the structure. This step is especially valuable when applying:", "- Factoring techniques — smaller coefficients make identifying pairs of factors faster.\n- The quadratic formula — accurate coefficients ensure precise computation.\n- Graphing and vertex analysis — the simplified ( ax^2 + bx + c ) clearly reveals key features like maximum/minimum points and axis of symmetry.", "---", "### Step-by-Step: Solving ( 2n^2 + 2n - 84 = 0 )", "Now that the equation is simplified, let’s solve it efficiently.", "Step 1: Factor out the greatest common factor (GCF)\nThe coefficients 2, 2, and -84 share a common factor of 2:", "[\n2(n^2 + n - 42) = 0\n]", "We now solve the simpler equation:", "[\nn^2 + n - 42 = 0\n]", "Step 2: Factor the quadratic\nWe look for two numbers that multiply to ( -42 ) and add to ( 1 ). These numbers are ( 7 ) and ( -6 ):", "[\n(n + 7)(n - 6) = 0\n]", "Step 3: Solve for ( n )\nSetting each factor equal to zero:", "[\nn + 7 = 0 \quad \Rightarrow \quad n = -7\n]\n[\nn - 6 = 0 \quad \Rightarrow \quad n = 6\n]", "---", "### Final Answer", "The simplified equation after subtracting 85 from both sides is:", "[\n2n^2 + 2n - 84 = 0\n]", "Solving this yields two solutions:\n[\nn = -7 \quad \ ext{and} \quad n = 6\n]", "---", "### SEO Keywords & Phrases\nOptimized for search engines:\n- How to simplify quadratic equations\n- Subtracting constants in algebra\n- Step-by-step solution to ( 2n^2 + 2n - 84 = 0 )\n- Quadratic equation solving techniques\n- Simplify quadratic expressions\n- Two-step quadratic equation tutorial", " related long-tail keywords:\n- Solving ( 2n^2 + 2n + 1 - 85 = 0 ) by subtracting 85\n- Complete guide to factoring quadratic equations\n- Best methods for solving ax² + bx + c = 0 quickly", "---", "Conclusion", "Subtracting 85 from both sides of the equation simplifies the expression from ( 2n^2 + 2n + 1 - 85 = 0 ) to ( 2n^2 + 2n - 84 = 0 ), streamlining the path to solution. This foundational algebra step enhances clarity, enabling faster and more accurate problem-solving—essential for mastering quadratic equations.", "For more math tips and detailed algebraic techniques, explore our full library of educational resources.", "---", "Keywords: quadratic equations, simplify algebra, solve 2n² + 2n - 84 = 0, subtract 85 from equation, step-by-step quadratic solution, algebra tutorial, factoring quadratics, quadratic formula tips"]
