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Simplifying 35n2 + n + 3 = 0 Reorder the terms: 3 + n + 35n2 = 0 Solving 3 + n + 35n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by 35 the coefficient of the squared term: Divide each side by '35'. 0.08571428571 + 0.02857142857n + n2 = 0 Move the constant term to the right: Add '-0.08571428571' to each side of the equation. 0.08571428571 + 0.02857142857n + -0.08571428571 + n2 = 0 + -0.08571428571 Reorder the terms: 0.08571428571 + -0.08571428571 + 0.02857142857n + n2 = 0 + -0.08571428571 Combine like terms: 0.08571428571 + -0.08571428571 = 0.00000000000 0.00000000000 + 0.02857142857n + n2 = 0 + -0.08571428571 0.02857142857n + n2 = 0 + -0.08571428571 Combine like terms: 0 + -0.08571428571 = -0.08571428571 0.02857142857n + n2 = -0.08571428571 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. 0.02857142857n + 0.25 + n2 = -0.08571428571 + 0.25 Reorder the terms: 0.25 + 0.02857142857n + n2 = -0.08571428571 + 0.25 Combine like terms: -0.08571428571 + 0.25 = 0.16428571429 0.25 + 0.02857142857n + n2 = 0.16428571429 Factor a perfect square on the left side: (n + 0.5)(n + 0.5) = 0.16428571429 Calculate the square root of the right side: 0.405321742 Break this problem into two subproblems by setting (n + 0.5) equal to 0.405321742 and -0.405321742.Subproblem 1
n + 0.5 = 0.405321742 Simplifying n + 0.5 = 0.405321742 Reorder the terms: 0.5 + n = 0.405321742 Solving 0.5 + n = 0.405321742 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 = 0.405321742 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 0.405321742 + -0.5 n = 0.405321742 + -0.5 Combine like terms: 0.405321742 + -0.5 = -0.094678258 n = -0.094678258 Simplifying n = -0.094678258Subproblem 2
n + 0.5 = -0.405321742 Simplifying n + 0.5 = -0.405321742 Reorder the terms: 0.5 + n = -0.405321742 Solving 0.5 + n = -0.405321742 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 = -0.405321742 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -0.405321742 + -0.5 n = -0.405321742 + -0.5 Combine like terms: -0.405321742 + -0.5 = -0.905321742 n = -0.905321742 Simplifying n = -0.905321742Solution
The solution to the problem is based on the solutions from the subproblems. n = {-0.094678258, -0.905321742}
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