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Simplifying 1 + n + 6n2 = 0 Solving 1 + n + 6n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. 0.1666666667 + 0.1666666667n + n2 = 0 Move the constant term to the right: Add '-0.1666666667' to each side of the equation. 0.1666666667 + 0.1666666667n + -0.1666666667 + n2 = 0 + -0.1666666667 Reorder the terms: 0.1666666667 + -0.1666666667 + 0.1666666667n + n2 = 0 + -0.1666666667 Combine like terms: 0.1666666667 + -0.1666666667 = 0.0000000000 0.0000000000 + 0.1666666667n + n2 = 0 + -0.1666666667 0.1666666667n + n2 = 0 + -0.1666666667 Combine like terms: 0 + -0.1666666667 = -0.1666666667 0.1666666667n + n2 = -0.1666666667 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. 0.1666666667n + 0.25 + n2 = -0.1666666667 + 0.25 Reorder the terms: 0.25 + 0.1666666667n + n2 = -0.1666666667 + 0.25 Combine like terms: -0.1666666667 + 0.25 = 0.0833333333 0.25 + 0.1666666667n + n2 = 0.0833333333 Factor a perfect square on the left side: (n + 0.5)(n + 0.5) = 0.0833333333 Calculate the square root of the right side: 0.288675135 Break this problem into two subproblems by setting (n + 0.5) equal to 0.288675135 and -0.288675135.Subproblem 1
n + 0.5 = 0.288675135 Simplifying n + 0.5 = 0.288675135 Reorder the terms: 0.5 + n = 0.288675135 Solving 0.5 + n = 0.288675135 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 = 0.288675135 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 0.288675135 + -0.5 n = 0.288675135 + -0.5 Combine like terms: 0.288675135 + -0.5 = -0.211324865 n = -0.211324865 Simplifying n = -0.211324865Subproblem 2
n + 0.5 = -0.288675135 Simplifying n + 0.5 = -0.288675135 Reorder the terms: 0.5 + n = -0.288675135 Solving 0.5 + n = -0.288675135 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 = -0.288675135 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -0.288675135 + -0.5 n = -0.288675135 + -0.5 Combine like terms: -0.288675135 + -0.5 = -0.788675135 n = -0.788675135 Simplifying n = -0.788675135Solution
The solution to the problem is based on the solutions from the subproblems. n = {-0.211324865, -0.788675135}
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