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Simplifying -4n2 + 6n + -16 = -15n2 Reorder the terms: -16 + 6n + -4n2 = -15n2 Solving -16 + 6n + -4n2 = -15n2 Solving for variable 'n'. Combine like terms: -4n2 + 15n2 = 11n2 -16 + 6n + 11n2 = -15n2 + 15n2 Combine like terms: -15n2 + 15n2 = 0 -16 + 6n + 11n2 = 0 Begin completing the square. Divide all terms by 11 the coefficient of the squared term: Divide each side by '11'. -1.454545455 + 0.5454545455n + n2 = 0 Move the constant term to the right: Add '1.454545455' to each side of the equation. -1.454545455 + 0.5454545455n + 1.454545455 + n2 = 0 + 1.454545455 Reorder the terms: -1.454545455 + 1.454545455 + 0.5454545455n + n2 = 0 + 1.454545455 Combine like terms: -1.454545455 + 1.454545455 = 0.000000000 0.000000000 + 0.5454545455n + n2 = 0 + 1.454545455 0.5454545455n + n2 = 0 + 1.454545455 Combine like terms: 0 + 1.454545455 = 1.454545455 0.5454545455n + n2 = 1.454545455 The n term is 0.5454545455n. Take half its coefficient (0.2727272728). Square it (0.07438016533) and add it to both sides. Add '0.07438016533' to each side of the equation. 0.5454545455n + 0.07438016533 + n2 = 1.454545455 + 0.07438016533 Reorder the terms: 0.07438016533 + 0.5454545455n + n2 = 1.454545455 + 0.07438016533 Combine like terms: 1.454545455 + 0.07438016533 = 1.52892562033 0.07438016533 + 0.5454545455n + n2 = 1.52892562033 Factor a perfect square on the left side: (n + 0.2727272728)(n + 0.2727272728) = 1.52892562033 Calculate the square root of the right side: 1.236497319 Break this problem into two subproblems by setting (n + 0.2727272728) equal to 1.236497319 and -1.236497319.Subproblem 1
n + 0.2727272728 = 1.236497319 Simplifying n + 0.2727272728 = 1.236497319 Reorder the terms: 0.2727272728 + n = 1.236497319 Solving 0.2727272728 + n = 1.236497319 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.2727272728' to each side of the equation. 0.2727272728 + -0.2727272728 + n = 1.236497319 + -0.2727272728 Combine like terms: 0.2727272728 + -0.2727272728 = 0.0000000000 0.0000000000 + n = 1.236497319 + -0.2727272728 n = 1.236497319 + -0.2727272728 Combine like terms: 1.236497319 + -0.2727272728 = 0.9637700462 n = 0.9637700462 Simplifying n = 0.9637700462Subproblem 2
n + 0.2727272728 = -1.236497319 Simplifying n + 0.2727272728 = -1.236497319 Reorder the terms: 0.2727272728 + n = -1.236497319 Solving 0.2727272728 + n = -1.236497319 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.2727272728' to each side of the equation. 0.2727272728 + -0.2727272728 + n = -1.236497319 + -0.2727272728 Combine like terms: 0.2727272728 + -0.2727272728 = 0.0000000000 0.0000000000 + n = -1.236497319 + -0.2727272728 n = -1.236497319 + -0.2727272728 Combine like terms: -1.236497319 + -0.2727272728 = -1.5092245918 n = -1.5092245918 Simplifying n = -1.5092245918Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.9637700462, -1.5092245918}
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