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Simplifying -6m + 16 + -6m(m + -1) = 4m + -1(2m + -1) + -8m + 9 Reorder the terms: -6m + 16 + -6m(-1 + m) = 4m + -1(2m + -1) + -8m + 9 -6m + 16 + (-1 * -6m + m * -6m) = 4m + -1(2m + -1) + -8m + 9 -6m + 16 + (6m + -6m2) = 4m + -1(2m + -1) + -8m + 9 Reorder the terms: 16 + -6m + 6m + -6m2 = 4m + -1(2m + -1) + -8m + 9 Combine like terms: -6m + 6m = 0 16 + 0 + -6m2 = 4m + -1(2m + -1) + -8m + 9 16 + -6m2 = 4m + -1(2m + -1) + -8m + 9 Reorder the terms: 16 + -6m2 = 4m + -1(-1 + 2m) + -8m + 9 16 + -6m2 = 4m + (-1 * -1 + 2m * -1) + -8m + 9 16 + -6m2 = 4m + (1 + -2m) + -8m + 9 Reorder the terms: 16 + -6m2 = 1 + 9 + 4m + -2m + -8m Combine like terms: 1 + 9 = 10 16 + -6m2 = 10 + 4m + -2m + -8m Combine like terms: 4m + -2m = 2m 16 + -6m2 = 10 + 2m + -8m Combine like terms: 2m + -8m = -6m 16 + -6m2 = 10 + -6m Solving 16 + -6m2 = 10 + -6m Solving for variable 'm'. Reorder the terms: 16 + -10 + 6m + -6m2 = 10 + -6m + -10 + 6m Combine like terms: 16 + -10 = 6 6 + 6m + -6m2 = 10 + -6m + -10 + 6m Reorder the terms: 6 + 6m + -6m2 = 10 + -10 + -6m + 6m Combine like terms: 10 + -10 = 0 6 + 6m + -6m2 = 0 + -6m + 6m 6 + 6m + -6m2 = -6m + 6m Combine like terms: -6m + 6m = 0 6 + 6m + -6m2 = 0 Factor out the Greatest Common Factor (GCF), '6'. 6(1 + m + -1m2) = 0 Ignore the factor 6.Subproblem 1
Set the factor '(1 + m + -1m2)' equal to zero and attempt to solve: Simplifying 1 + m + -1m2 = 0 Solving 1 + m + -1m2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -1 + -1m + m2 = 0 Move the constant term to the right: Add '1' to each side of the equation. -1 + -1m + 1 + m2 = 0 + 1 Reorder the terms: -1 + 1 + -1m + m2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + -1m + m2 = 0 + 1 -1m + m2 = 0 + 1 Combine like terms: 0 + 1 = 1 -1m + m2 = 1 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. -1m + 0.25 + m2 = 1 + 0.25 Reorder the terms: 0.25 + -1m + m2 = 1 + 0.25 Combine like terms: 1 + 0.25 = 1.25 0.25 + -1m + m2 = 1.25 Factor a perfect square on the left side: (m + 0.5)(m + 0.5) = 1.25 Calculate the square root of the right side: 1.118033989 Break this problem into two subproblems by setting (m + 0.5) equal to 1.118033989 and -1.118033989.Subproblem 1
m + 0.5 = 1.118033989 Simplifying m + 0.5 = 1.118033989 Reorder the terms: 0.5 + m = 1.118033989 Solving 0.5 + m = 1.118033989 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 = 1.118033989 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + m = 1.118033989 + -0.5 m = 1.118033989 + -0.5 Combine like terms: 1.118033989 + -0.5 = 0.618033989 m = 0.618033989 Simplifying m = 0.618033989Subproblem 2
m + 0.5 = -1.118033989 Simplifying m + 0.5 = -1.118033989 Reorder the terms: 0.5 + m = -1.118033989 Solving 0.5 + m = -1.118033989 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 = -1.118033989 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + m = -1.118033989 + -0.5 m = -1.118033989 + -0.5 Combine like terms: -1.118033989 + -0.5 = -1.618033989 m = -1.618033989 Simplifying m = -1.618033989Solution
The solution to the problem is based on the solutions from the subproblems. m = {0.618033989, -1.618033989}Solution
m = {0.618033989, -1.618033989}
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