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Simplifying n2 + 4n + 3 = 2016 Reorder the terms: 3 + 4n + n2 = 2016 Solving 3 + 4n + n2 = 2016 Solving for variable 'n'. Reorder the terms: 3 + -2016 + 4n + n2 = 2016 + -2016 Combine like terms: 3 + -2016 = -2013 -2013 + 4n + n2 = 2016 + -2016 Combine like terms: 2016 + -2016 = 0 -2013 + 4n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '2013' to each side of the equation. -2013 + 4n + 2013 + n2 = 0 + 2013 Reorder the terms: -2013 + 2013 + 4n + n2 = 0 + 2013 Combine like terms: -2013 + 2013 = 0 0 + 4n + n2 = 0 + 2013 4n + n2 = 0 + 2013 Combine like terms: 0 + 2013 = 2013 4n + n2 = 2013 The n term is 4n. Take half its coefficient (2). Square it (4) and add it to both sides. Add '4' to each side of the equation. 4n + 4 + n2 = 2013 + 4 Reorder the terms: 4 + 4n + n2 = 2013 + 4 Combine like terms: 2013 + 4 = 2017 4 + 4n + n2 = 2017 Factor a perfect square on the left side: (n + 2)(n + 2) = 2017 Calculate the square root of the right side: 44.911023146 Break this problem into two subproblems by setting (n + 2) equal to 44.911023146 and -44.911023146.Subproblem 1
n + 2 = 44.911023146 Simplifying n + 2 = 44.911023146 Reorder the terms: 2 + n = 44.911023146 Solving 2 + n = 44.911023146 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + n = 44.911023146 + -2 Combine like terms: 2 + -2 = 0 0 + n = 44.911023146 + -2 n = 44.911023146 + -2 Combine like terms: 44.911023146 + -2 = 42.911023146 n = 42.911023146 Simplifying n = 42.911023146Subproblem 2
n + 2 = -44.911023146 Simplifying n + 2 = -44.911023146 Reorder the terms: 2 + n = -44.911023146 Solving 2 + n = -44.911023146 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + n = -44.911023146 + -2 Combine like terms: 2 + -2 = 0 0 + n = -44.911023146 + -2 n = -44.911023146 + -2 Combine like terms: -44.911023146 + -2 = -46.911023146 n = -46.911023146 Simplifying n = -46.911023146Solution
The solution to the problem is based on the solutions from the subproblems. n = {42.911023146, -46.911023146}
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