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Simplifying 4n2 + -8n + 1 = 0 Reorder the terms: 1 + -8n + 4n2 = 0 Solving 1 + -8n + 4n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. 0.25 + -2n + n2 = 0 Move the constant term to the right: Add '-0.25' to each side of the equation. 0.25 + -2n + -0.25 + n2 = 0 + -0.25 Reorder the terms: 0.25 + -0.25 + -2n + n2 = 0 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + -2n + n2 = 0 + -0.25 -2n + n2 = 0 + -0.25 Combine like terms: 0 + -0.25 = -0.25 -2n + n2 = -0.25 The n term is -2n. Take half its coefficient (-1). Square it (1) and add it to both sides. Add '1' to each side of the equation. -2n + 1 + n2 = -0.25 + 1 Reorder the terms: 1 + -2n + n2 = -0.25 + 1 Combine like terms: -0.25 + 1 = 0.75 1 + -2n + n2 = 0.75 Factor a perfect square on the left side: (n + -1)(n + -1) = 0.75 Calculate the square root of the right side: 0.866025404 Break this problem into two subproblems by setting (n + -1) equal to 0.866025404 and -0.866025404.Subproblem 1
n + -1 = 0.866025404 Simplifying n + -1 = 0.866025404 Reorder the terms: -1 + n = 0.866025404 Solving -1 + n = 0.866025404 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + n = 0.866025404 + 1 Combine like terms: -1 + 1 = 0 0 + n = 0.866025404 + 1 n = 0.866025404 + 1 Combine like terms: 0.866025404 + 1 = 1.866025404 n = 1.866025404 Simplifying n = 1.866025404Subproblem 2
n + -1 = -0.866025404 Simplifying n + -1 = -0.866025404 Reorder the terms: -1 + n = -0.866025404 Solving -1 + n = -0.866025404 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + n = -0.866025404 + 1 Combine like terms: -1 + 1 = 0 0 + n = -0.866025404 + 1 n = -0.866025404 + 1 Combine like terms: -0.866025404 + 1 = 0.133974596 n = 0.133974596 Simplifying n = 0.133974596Solution
The solution to the problem is based on the solutions from the subproblems. n = {1.866025404, 0.133974596}
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