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Simplifying n2 + -45n = 116 Reorder the terms: -45n + n2 = 116 Solving -45n + n2 = 116 Solving for variable 'n'. Reorder the terms: -116 + -45n + n2 = 116 + -116 Combine like terms: 116 + -116 = 0 -116 + -45n + n2 = 0 Begin completing the square. Move the constant term to the right: Add '116' to each side of the equation. -116 + -45n + 116 + n2 = 0 + 116 Reorder the terms: -116 + 116 + -45n + n2 = 0 + 116 Combine like terms: -116 + 116 = 0 0 + -45n + n2 = 0 + 116 -45n + n2 = 0 + 116 Combine like terms: 0 + 116 = 116 -45n + n2 = 116 The n term is -45n. Take half its coefficient (-22.5). Square it (506.25) and add it to both sides. Add '506.25' to each side of the equation. -45n + 506.25 + n2 = 116 + 506.25 Reorder the terms: 506.25 + -45n + n2 = 116 + 506.25 Combine like terms: 116 + 506.25 = 622.25 506.25 + -45n + n2 = 622.25 Factor a perfect square on the left side: (n + -22.5)(n + -22.5) = 622.25 Calculate the square root of the right side: 24.944939367 Break this problem into two subproblems by setting (n + -22.5) equal to 24.944939367 and -24.944939367.Subproblem 1
n + -22.5 = 24.944939367 Simplifying n + -22.5 = 24.944939367 Reorder the terms: -22.5 + n = 24.944939367 Solving -22.5 + n = 24.944939367 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '22.5' to each side of the equation. -22.5 + 22.5 + n = 24.944939367 + 22.5 Combine like terms: -22.5 + 22.5 = 0.0 0.0 + n = 24.944939367 + 22.5 n = 24.944939367 + 22.5 Combine like terms: 24.944939367 + 22.5 = 47.444939367 n = 47.444939367 Simplifying n = 47.444939367Subproblem 2
n + -22.5 = -24.944939367 Simplifying n + -22.5 = -24.944939367 Reorder the terms: -22.5 + n = -24.944939367 Solving -22.5 + n = -24.944939367 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '22.5' to each side of the equation. -22.5 + 22.5 + n = -24.944939367 + 22.5 Combine like terms: -22.5 + 22.5 = 0.0 0.0 + n = -24.944939367 + 22.5 n = -24.944939367 + 22.5 Combine like terms: -24.944939367 + 22.5 = -2.444939367 n = -2.444939367 Simplifying n = -2.444939367Solution
The solution to the problem is based on the solutions from the subproblems. n = {47.444939367, -2.444939367}
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