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Simplifying 30n2 + n2 + n = 1 Reorder the terms: n + 30n2 + n2 = 1 Combine like terms: 30n2 + n2 = 31n2 n + 31n2 = 1 Solving n + 31n2 = 1 Solving for variable 'n'. Reorder the terms: -1 + n + 31n2 = 1 + -1 Combine like terms: 1 + -1 = 0 -1 + n + 31n2 = 0 Begin completing the square. Divide all terms by 31 the coefficient of the squared term: Divide each side by '31'. -0.03225806452 + 0.03225806452n + n2 = 0 Move the constant term to the right: Add '0.03225806452' to each side of the equation. -0.03225806452 + 0.03225806452n + 0.03225806452 + n2 = 0 + 0.03225806452 Reorder the terms: -0.03225806452 + 0.03225806452 + 0.03225806452n + n2 = 0 + 0.03225806452 Combine like terms: -0.03225806452 + 0.03225806452 = 0.00000000000 0.00000000000 + 0.03225806452n + n2 = 0 + 0.03225806452 0.03225806452n + n2 = 0 + 0.03225806452 Combine like terms: 0 + 0.03225806452 = 0.03225806452 0.03225806452n + n2 = 0.03225806452 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. 0.03225806452n + 0.25 + n2 = 0.03225806452 + 0.25 Reorder the terms: 0.25 + 0.03225806452n + n2 = 0.03225806452 + 0.25 Combine like terms: 0.03225806452 + 0.25 = 0.28225806452 0.25 + 0.03225806452n + n2 = 0.28225806452 Factor a perfect square on the left side: (n + 0.5)(n + 0.5) = 0.28225806452 Calculate the square root of the right side: 0.531279648 Break this problem into two subproblems by setting (n + 0.5) equal to 0.531279648 and -0.531279648.Subproblem 1
n + 0.5 = 0.531279648 Simplifying n + 0.5 = 0.531279648 Reorder the terms: 0.5 + n = 0.531279648 Solving 0.5 + n = 0.531279648 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 = 0.531279648 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 0.531279648 + -0.5 n = 0.531279648 + -0.5 Combine like terms: 0.531279648 + -0.5 = 0.031279648 n = 0.031279648 Simplifying n = 0.031279648Subproblem 2
n + 0.5 = -0.531279648 Simplifying n + 0.5 = -0.531279648 Reorder the terms: 0.5 + n = -0.531279648 Solving 0.5 + n = -0.531279648 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 = -0.531279648 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -0.531279648 + -0.5 n = -0.531279648 + -0.5 Combine like terms: -0.531279648 + -0.5 = -1.031279648 n = -1.031279648 Simplifying n = -1.031279648Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.031279648, -1.031279648}
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