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Simplifying 83 = 0.25M2 + -1M + 68 Reorder the terms: 83 = 68 + -1M + 0.25M2 Solving 83 = 68 + -1M + 0.25M2 Solving for variable 'M'. Combine like terms: 83 + -68 = 15 15 + M + -0.25M2 = 68 + -1M + 0.25M2 + -68 + M + -0.25M2 Reorder the terms: 15 + M + -0.25M2 = 68 + -68 + -1M + M + 0.25M2 + -0.25M2 Combine like terms: 68 + -68 = 0 15 + M + -0.25M2 = 0 + -1M + M + 0.25M2 + -0.25M2 15 + M + -0.25M2 = -1M + M + 0.25M2 + -0.25M2 Combine like terms: -1M + M = 0 15 + M + -0.25M2 = 0 + 0.25M2 + -0.25M2 15 + M + -0.25M2 = 0.25M2 + -0.25M2 Combine like terms: 0.25M2 + -0.25M2 = 0.00 15 + M + -0.25M2 = 0.00 Begin completing the square. Divide all terms by -0.25 the coefficient of the squared term: Divide each side by '-0.25'. -60 + -4M + M2 = 0 Move the constant term to the right: Add '60' to each side of the equation. -60 + -4M + 60 + M2 = 0 + 60 Reorder the terms: -60 + 60 + -4M + M2 = 0 + 60 Combine like terms: -60 + 60 = 0 0 + -4M + M2 = 0 + 60 -4M + M2 = 0 + 60 Combine like terms: 0 + 60 = 60 -4M + M2 = 60 The M term is M. 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. -4M + 0.25 + M2 = 60 + 0.25 Reorder the terms: 0.25 + -4M + M2 = 60 + 0.25 Combine like terms: 60 + 0.25 = 60.25 0.25 + -4M + M2 = 60.25 Factor a perfect square on the left side: (M + 0.5)(M + 0.5) = 60.25 Calculate the square root of the right side: 7.762087348 Break this problem into two subproblems by setting (M + 0.5) equal to 7.762087348 and -7.762087348.Subproblem 1
M + 0.5 = 7.762087348 Simplifying M + 0.5 = 7.762087348 Reorder the terms: 0.5 + M = 7.762087348 Solving 0.5 + M = 7.762087348 Solving for variable 'M'. Move all terms containing M to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + M = 7.762087348 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + M = 7.762087348 + -0.5 M = 7.762087348 + -0.5 Combine like terms: 7.762087348 + -0.5 = 7.262087348 M = 7.262087348 Simplifying M = 7.262087348Subproblem 2
M + 0.5 = -7.762087348 Simplifying M + 0.5 = -7.762087348 Reorder the terms: 0.5 + M = -7.762087348 Solving 0.5 + M = -7.762087348 Solving for variable 'M'. Move all terms containing M to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + M = -7.762087348 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + M = -7.762087348 + -0.5 M = -7.762087348 + -0.5 Combine like terms: -7.762087348 + -0.5 = -8.262087348 M = -8.262087348 Simplifying M = -8.262087348Solution
The solution to the problem is based on the solutions from the subproblems. M = {7.262087348, -8.262087348}
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