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Simplifying n2 + -1n = 32 Reorder the terms: -1n + n2 = 32 Solving -1n + n2 = 32 Solving for variable 'n'. Reorder the terms: -32 + -1n + n2 = 32 + -32 Combine like terms: 32 + -32 = 0 -32 + -1n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '32' to each side of the equation. -32 + -1n + 32 + n2 = 0 + 32 Reorder the terms: -32 + 32 + -1n + n2 = 0 + 32 Combine like terms: -32 + 32 = 0 0 + -1n + n2 = 0 + 32 -1n + n2 = 0 + 32 Combine like terms: 0 + 32 = 32 -1n + n2 = 32 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 = 32 + 0.25 Reorder the terms: 0.25 + -1n + n2 = 32 + 0.25 Combine like terms: 32 + 0.25 = 32.25 0.25 + -1n + n2 = 32.25 Factor a perfect square on the left side: (n + -0.5)(n + -0.5) = 32.25 Calculate the square root of the right side: 5.678908346 Break this problem into two subproblems by setting (n + -0.5) equal to 5.678908346 and -5.678908346.Subproblem 1
n + -0.5 = 5.678908346 Simplifying n + -0.5 = 5.678908346 Reorder the terms: -0.5 + n = 5.678908346 Solving -0.5 + n = 5.678908346 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 = 5.678908346 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = 5.678908346 + 0.5 n = 5.678908346 + 0.5 Combine like terms: 5.678908346 + 0.5 = 6.178908346 n = 6.178908346 Simplifying n = 6.178908346Subproblem 2
n + -0.5 = -5.678908346 Simplifying n + -0.5 = -5.678908346 Reorder the terms: -0.5 + n = -5.678908346 Solving -0.5 + n = -5.678908346 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 = -5.678908346 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = -5.678908346 + 0.5 n = -5.678908346 + 0.5 Combine like terms: -5.678908346 + 0.5 = -5.178908346 n = -5.178908346 Simplifying n = -5.178908346Solution
The solution to the problem is based on the solutions from the subproblems. n = {6.178908346, -5.178908346}
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