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Simplifying 12n2 + 12n = -1 Reorder the terms: 12n + 12n2 = -1 Solving 12n + 12n2 = -1 Solving for variable 'n'. Reorder the terms: 1 + 12n + 12n2 = -1 + 1 Combine like terms: -1 + 1 = 0 1 + 12n + 12n2 = 0 Begin completing the square. Divide all terms by 12 the coefficient of the squared term: Divide each side by '12'. 0.08333333333 + n + n2 = 0 Move the constant term to the right: Add '-0.08333333333' to each side of the equation. 0.08333333333 + n + -0.08333333333 + n2 = 0 + -0.08333333333 Reorder the terms: 0.08333333333 + -0.08333333333 + n + n2 = 0 + -0.08333333333 Combine like terms: 0.08333333333 + -0.08333333333 = 0.00000000000 0.00000000000 + n + n2 = 0 + -0.08333333333 n + n2 = 0 + -0.08333333333 Combine like terms: 0 + -0.08333333333 = -0.08333333333 n + n2 = -0.08333333333 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. + 0.25 + n2 = -0.08333333333 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + n2 = -0.08333333333 + 0.25 Combine like terms: -0.08333333333 + 0.25 = 0.16666666667 1.25 + n2 = 0.16666666667 Factor a perfect square on the left side: (n + 0.5)(n + 0.5) = 0.16666666667 Calculate the square root of the right side: 0.40824829 Break this problem into two subproblems by setting (n + 0.5) equal to 0.40824829 and -0.40824829.Subproblem 1
n + 0.5 = 0.40824829 Simplifying n + 0.5 = 0.40824829 Reorder the terms: 0.5 + n = 0.40824829 Solving 0.5 + n = 0.40824829 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 = 0.40824829 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 0.40824829 + -0.5 n = 0.40824829 + -0.5 Combine like terms: 0.40824829 + -0.5 = -0.09175171 n = -0.09175171 Simplifying n = -0.09175171Subproblem 2
n + 0.5 = -0.40824829 Simplifying n + 0.5 = -0.40824829 Reorder the terms: 0.5 + n = -0.40824829 Solving 0.5 + n = -0.40824829 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 = -0.40824829 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -0.40824829 + -0.5 n = -0.40824829 + -0.5 Combine like terms: -0.40824829 + -0.5 = -0.90824829 n = -0.90824829 Simplifying n = -0.90824829Solution
The solution to the problem is based on the solutions from the subproblems. n = {-0.09175171, -0.90824829}
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