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Simplifying 4m(6m + 5) + 2 = 8m + 3(4 + -3m) Reorder the terms: 4m(5 + 6m) + 2 = 8m + 3(4 + -3m) (5 * 4m + 6m * 4m) + 2 = 8m + 3(4 + -3m) (20m + 24m2) + 2 = 8m + 3(4 + -3m) Reorder the terms: 2 + 20m + 24m2 = 8m + 3(4 + -3m) 2 + 20m + 24m2 = 8m + (4 * 3 + -3m * 3) 2 + 20m + 24m2 = 8m + (12 + -9m) Reorder the terms: 2 + 20m + 24m2 = 12 + 8m + -9m Combine like terms: 8m + -9m = -1m 2 + 20m + 24m2 = 12 + -1m Solving 2 + 20m + 24m2 = 12 + -1m Solving for variable 'm'. Reorder the terms: 2 + -12 + 20m + m + 24m2 = 12 + -1m + -12 + m Combine like terms: 2 + -12 = -10 -10 + 20m + m + 24m2 = 12 + -1m + -12 + m Combine like terms: 20m + m = 21m -10 + 21m + 24m2 = 12 + -1m + -12 + m Reorder the terms: -10 + 21m + 24m2 = 12 + -12 + -1m + m Combine like terms: 12 + -12 = 0 -10 + 21m + 24m2 = 0 + -1m + m -10 + 21m + 24m2 = -1m + m Combine like terms: -1m + m = 0 -10 + 21m + 24m2 = 0 Begin completing the square. Divide all terms by 24 the coefficient of the squared term: Divide each side by '24'. -0.4166666667 + 0.875m + m2 = 0 Move the constant term to the right: Add '0.4166666667' to each side of the equation. -0.4166666667 + 0.875m + 0.4166666667 + m2 = 0 + 0.4166666667 Reorder the terms: -0.4166666667 + 0.4166666667 + 0.875m + m2 = 0 + 0.4166666667 Combine like terms: -0.4166666667 + 0.4166666667 = 0.0000000000 0.0000000000 + 0.875m + m2 = 0 + 0.4166666667 0.875m + m2 = 0 + 0.4166666667 Combine like terms: 0 + 0.4166666667 = 0.4166666667 0.875m + m2 = 0.4166666667 The m term is 0.875m. Take half its coefficient (0.4375). Square it (0.19140625) and add it to both sides. Add '0.19140625' to each side of the equation. 0.875m + 0.19140625 + m2 = 0.4166666667 + 0.19140625 Reorder the terms: 0.19140625 + 0.875m + m2 = 0.4166666667 + 0.19140625 Combine like terms: 0.4166666667 + 0.19140625 = 0.6080729167 0.19140625 + 0.875m + m2 = 0.6080729167 Factor a perfect square on the left side: (m + 0.4375)(m + 0.4375) = 0.6080729167 Calculate the square root of the right side: 0.779790303 Break this problem into two subproblems by setting (m + 0.4375) equal to 0.779790303 and -0.779790303.Subproblem 1
m + 0.4375 = 0.779790303 Simplifying m + 0.4375 = 0.779790303 Reorder the terms: 0.4375 + m = 0.779790303 Solving 0.4375 + m = 0.779790303 Solving for variable 'm'. Move all terms containing m to the left, all other terms to the right. Add '-0.4375' to each side of the equation. 0.4375 + -0.4375 + m = 0.779790303 + -0.4375 Combine like terms: 0.4375 + -0.4375 = 0.0000 0.0000 + m = 0.779790303 + -0.4375 m = 0.779790303 + -0.4375 Combine like terms: 0.779790303 + -0.4375 = 0.342290303 m = 0.342290303 Simplifying m = 0.342290303Subproblem 2
m + 0.4375 = -0.779790303 Simplifying m + 0.4375 = -0.779790303 Reorder the terms: 0.4375 + m = -0.779790303 Solving 0.4375 + m = -0.779790303 Solving for variable 'm'. Move all terms containing m to the left, all other terms to the right. Add '-0.4375' to each side of the equation. 0.4375 + -0.4375 + m = -0.779790303 + -0.4375 Combine like terms: 0.4375 + -0.4375 = 0.0000 0.0000 + m = -0.779790303 + -0.4375 m = -0.779790303 + -0.4375 Combine like terms: -0.779790303 + -0.4375 = -1.217290303 m = -1.217290303 Simplifying m = -1.217290303Solution
The solution to the problem is based on the solutions from the subproblems. m = {0.342290303, -1.217290303}
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