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Simplifying n2 + -1n + -20 = 45 Reorder the terms: -20 + -1n + n2 = 45 Solving -20 + -1n + n2 = 45 Solving for variable 'n'. Reorder the terms: -20 + -45 + -1n + n2 = 45 + -45 Combine like terms: -20 + -45 = -65 -65 + -1n + n2 = 45 + -45 Combine like terms: 45 + -45 = 0 -65 + -1n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '65' to each side of the equation. -65 + -1n + 65 + n2 = 0 + 65 Reorder the terms: -65 + 65 + -1n + n2 = 0 + 65 Combine like terms: -65 + 65 = 0 0 + -1n + n2 = 0 + 65 -1n + n2 = 0 + 65 Combine like terms: 0 + 65 = 65 -1n + n2 = 65 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 = 65 + 0.25 Reorder the terms: 0.25 + -1n + n2 = 65 + 0.25 Combine like terms: 65 + 0.25 = 65.25 0.25 + -1n + n2 = 65.25 Factor a perfect square on the left side: (n + -0.5)(n + -0.5) = 65.25 Calculate the square root of the right side: 8.077747211 Break this problem into two subproblems by setting (n + -0.5) equal to 8.077747211 and -8.077747211.Subproblem 1
n + -0.5 = 8.077747211 Simplifying n + -0.5 = 8.077747211 Reorder the terms: -0.5 + n = 8.077747211 Solving -0.5 + n = 8.077747211 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 = 8.077747211 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = 8.077747211 + 0.5 n = 8.077747211 + 0.5 Combine like terms: 8.077747211 + 0.5 = 8.577747211 n = 8.577747211 Simplifying n = 8.577747211Subproblem 2
n + -0.5 = -8.077747211 Simplifying n + -0.5 = -8.077747211 Reorder the terms: -0.5 + n = -8.077747211 Solving -0.5 + n = -8.077747211 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 = -8.077747211 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + n = -8.077747211 + 0.5 n = -8.077747211 + 0.5 Combine like terms: -8.077747211 + 0.5 = -7.577747211 n = -7.577747211 Simplifying n = -7.577747211Solution
The solution to the problem is based on the solutions from the subproblems. n = {8.577747211, -7.577747211}
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