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Simplifying 2m2 + 6m + 3 = 0 Reorder the terms: 3 + 6m + 2m2 = 0 Solving 3 + 6m + 2m2 = 0 Solving for variable 'm'. Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 1.5 + 3m + m2 = 0 Move the constant term to the right: Add '-1.5' to each side of the equation. 1.5 + 3m + -1.5 + m2 = 0 + -1.5 Reorder the terms: 1.5 + -1.5 + 3m + m2 = 0 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + 3m + m2 = 0 + -1.5 3m + m2 = 0 + -1.5 Combine like terms: 0 + -1.5 = -1.5 3m + m2 = -1.5 The m term is 3m. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3m + 2.25 + m2 = -1.5 + 2.25 Reorder the terms: 2.25 + 3m + m2 = -1.5 + 2.25 Combine like terms: -1.5 + 2.25 = 0.75 2.25 + 3m + m2 = 0.75 Factor a perfect square on the left side: (m + 1.5)(m + 1.5) = 0.75 Calculate the square root of the right side: 0.866025404 Break this problem into two subproblems by setting (m + 1.5) equal to 0.866025404 and -0.866025404.Subproblem 1
m + 1.5 = 0.866025404 Simplifying m + 1.5 = 0.866025404 Reorder the terms: 1.5 + m = 0.866025404 Solving 1.5 + m = 0.866025404 Solving for variable 'm'. Move all terms containing m to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + m = 0.866025404 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + m = 0.866025404 + -1.5 m = 0.866025404 + -1.5 Combine like terms: 0.866025404 + -1.5 = -0.633974596 m = -0.633974596 Simplifying m = -0.633974596Subproblem 2
m + 1.5 = -0.866025404 Simplifying m + 1.5 = -0.866025404 Reorder the terms: 1.5 + m = -0.866025404 Solving 1.5 + m = -0.866025404 Solving for variable 'm'. Move all terms containing m to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + m = -0.866025404 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + m = -0.866025404 + -1.5 m = -0.866025404 + -1.5 Combine like terms: -0.866025404 + -1.5 = -2.366025404 m = -2.366025404 Simplifying m = -2.366025404Solution
The solution to the problem is based on the solutions from the subproblems. m = {-0.633974596, -2.366025404}
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