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Simplifying 12n2 = -1n + 16 Reorder the terms: 12n2 = 16 + -1n Solving 12n2 = 16 + -1n Solving for variable 'n'. Reorder the terms: -16 + n + 12n2 = 16 + -1n + -16 + n Reorder the terms: -16 + n + 12n2 = 16 + -16 + -1n + n Combine like terms: 16 + -16 = 0 -16 + n + 12n2 = 0 + -1n + n -16 + n + 12n2 = -1n + n Combine like terms: -1n + n = 0 -16 + n + 12n2 = 0 Begin completing the square. Divide all terms by 12 the coefficient of the squared term: Divide each side by '12'. -1.333333333 + 0.08333333333n + n2 = 0 Move the constant term to the right: Add '1.333333333' to each side of the equation. -1.333333333 + 0.08333333333n + 1.333333333 + n2 = 0 + 1.333333333 Reorder the terms: -1.333333333 + 1.333333333 + 0.08333333333n + n2 = 0 + 1.333333333 Combine like terms: -1.333333333 + 1.333333333 = 0.000000000 0.000000000 + 0.08333333333n + n2 = 0 + 1.333333333 0.08333333333n + n2 = 0 + 1.333333333 Combine like terms: 0 + 1.333333333 = 1.333333333 0.08333333333n + n2 = 1.333333333 The n term is n. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. 0.08333333333n + 0.25 + n2 = 1.333333333 + 0.25 Reorder the terms: 0.25 + 0.08333333333n + n2 = 1.333333333 + 0.25 Combine like terms: 1.333333333 + 0.25 = 1.583333333 0.25 + 0.08333333333n + n2 = 1.583333333 Factor a perfect square on the left side: (n + 0.5)(n + 0.5) = 1.583333333 Calculate the square root of the right side: 1.258305739 Break this problem into two subproblems by setting (n + 0.5) equal to 1.258305739 and -1.258305739.Subproblem 1
n + 0.5 = 1.258305739 Simplifying n + 0.5 = 1.258305739 Reorder the terms: 0.5 + n = 1.258305739 Solving 0.5 + n = 1.258305739 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + n = 1.258305739 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 1.258305739 + -0.5 n = 1.258305739 + -0.5 Combine like terms: 1.258305739 + -0.5 = 0.758305739 n = 0.758305739 Simplifying n = 0.758305739Subproblem 2
n + 0.5 = -1.258305739 Simplifying n + 0.5 = -1.258305739 Reorder the terms: 0.5 + n = -1.258305739 Solving 0.5 + n = -1.258305739 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + n = -1.258305739 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -1.258305739 + -0.5 n = -1.258305739 + -0.5 Combine like terms: -1.258305739 + -0.5 = -1.758305739 n = -1.758305739 Simplifying n = -1.758305739Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.758305739, -1.758305739}
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