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Simplifying 0.5n2 + -0.5n + -128 = 0 Reorder the terms: -128 + -0.5n + 0.5n2 = 0 Solving -128 + -0.5n + 0.5n2 = 0 Solving for variable 'n'. Begin completing the square. Divide all terms by 0.5 the coefficient of the squared term: Divide each side by '0.5'. -256 + -1n + n2 = 0 Move the constant term to the right: Add '256' to each side of the equation. -256 + -1n + 256 + n2 = 0 + 256 Reorder the terms: -256 + 256 + -1n + n2 = 0 + 256 Combine like terms: -256 + 256 = 0 0 + -1n + n2 = 0 + 256 -1n + n2 = 0 + 256 Combine like terms: 0 + 256 = 256 -1n + n2 = 256 The n term is -1n. 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. -1n + 0.25 + n2 = 256 + 0.25 Reorder the terms: 0.25 + -1n + n2 = 256 + 0.25 Combine like terms: 256 + 0.25 = 256.25 0.25 + -1n + n2 = 256.25 Factor a perfect square on the left side: (n + -0.5)(n + -0.5) = 256.25 Calculate the square root of the right side: 16.007810594 Break this problem into two subproblems by setting (n + -0.5) equal to 16.007810594 and -16.007810594.Subproblem 1
n + -0.5 = 16.007810594 Simplifying n + -0.5 = 16.007810594 Reorder the terms: -0.5 + n = 16.007810594 Solving -0.5 + n = 16.007810594 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 = 16.007810594 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = 16.007810594 + 0.5 n = 16.007810594 + 0.5 Combine like terms: 16.007810594 + 0.5 = 16.507810594 n = 16.507810594 Simplifying n = 16.507810594Subproblem 2
n + -0.5 = -16.007810594 Simplifying n + -0.5 = -16.007810594 Reorder the terms: -0.5 + n = -16.007810594 Solving -0.5 + n = -16.007810594 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 = -16.007810594 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = -16.007810594 + 0.5 n = -16.007810594 + 0.5 Combine like terms: -16.007810594 + 0.5 = -15.507810594 n = -15.507810594 Simplifying n = -15.507810594Solution
The solution to the problem is based on the solutions from the subproblems. n = {16.507810594, -15.507810594}
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