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Simplifying 81n2 + 18n + -1 = 0 Reorder the terms: -1 + 18n + 81n2 = 0 Solving -1 + 18n + 81n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by 81 the coefficient of the squared term: Divide each side by '81'. -0.01234567901 + 0.2222222222n + n2 = 0 Move the constant term to the right: Add '0.01234567901' to each side of the equation. -0.01234567901 + 0.2222222222n + 0.01234567901 + n2 = 0 + 0.01234567901 Reorder the terms: -0.01234567901 + 0.01234567901 + 0.2222222222n + n2 = 0 + 0.01234567901 Combine like terms: -0.01234567901 + 0.01234567901 = 0.00000000000 0.00000000000 + 0.2222222222n + n2 = 0 + 0.01234567901 0.2222222222n + n2 = 0 + 0.01234567901 Combine like terms: 0 + 0.01234567901 = 0.01234567901 0.2222222222n + n2 = 0.01234567901 The n term is 0.2222222222n. Take half its coefficient (0.1111111111). Square it (0.01234567901) and add it to both sides. Add '0.01234567901' to each side of the equation. 0.2222222222n + 0.01234567901 + n2 = 0.01234567901 + 0.01234567901 Reorder the terms: 0.01234567901 + 0.2222222222n + n2 = 0.01234567901 + 0.01234567901 Combine like terms: 0.01234567901 + 0.01234567901 = 0.02469135802 0.01234567901 + 0.2222222222n + n2 = 0.02469135802 Factor a perfect square on the left side: (n + 0.1111111111)(n + 0.1111111111) = 0.02469135802 Calculate the square root of the right side: 0.15713484 Break this problem into two subproblems by setting (n + 0.1111111111) equal to 0.15713484 and -0.15713484.Subproblem 1
n + 0.1111111111 = 0.15713484 Simplifying n + 0.1111111111 = 0.15713484 Reorder the terms: 0.1111111111 + n = 0.15713484 Solving 0.1111111111 + n = 0.15713484 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.1111111111' to each side of the equation. 0.1111111111 + -0.1111111111 + n = 0.15713484 + -0.1111111111 Combine like terms: 0.1111111111 + -0.1111111111 = 0.0000000000 0.0000000000 + n = 0.15713484 + -0.1111111111 n = 0.15713484 + -0.1111111111 Combine like terms: 0.15713484 + -0.1111111111 = 0.0460237289 n = 0.0460237289 Simplifying n = 0.0460237289Subproblem 2
n + 0.1111111111 = -0.15713484 Simplifying n + 0.1111111111 = -0.15713484 Reorder the terms: 0.1111111111 + n = -0.15713484 Solving 0.1111111111 + n = -0.15713484 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.1111111111' to each side of the equation. 0.1111111111 + -0.1111111111 + n = -0.15713484 + -0.1111111111 Combine like terms: 0.1111111111 + -0.1111111111 = 0.0000000000 0.0000000000 + n = -0.15713484 + -0.1111111111 n = -0.15713484 + -0.1111111111 Combine like terms: -0.15713484 + -0.1111111111 = -0.2682459511 n = -0.2682459511 Simplifying n = -0.2682459511Solution
The solution to the problem is based on the solutions from the subproblems. n = {0.0460237289, -0.2682459511}
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