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Simplifying 6(4m + 5) + 7m(2m + 3) = 0 Reorder the terms: 6(5 + 4m) + 7m(2m + 3) = 0 (5 * 6 + 4m * 6) + 7m(2m + 3) = 0 (30 + 24m) + 7m(2m + 3) = 0 Reorder the terms: 30 + 24m + 7m(3 + 2m) = 0 30 + 24m + (3 * 7m + 2m * 7m) = 0 30 + 24m + (21m + 14m2) = 0 Combine like terms: 24m + 21m = 45m 30 + 45m + 14m2 = 0 Solving 30 + 45m + 14m2 = 0 Solving for variable 'm'. Begin completing the square. Divide all terms by 14 the coefficient of the squared term: Divide each side by '14'. 2.142857143 + 3.214285714m + m2 = 0 Move the constant term to the right: Add '-2.142857143' to each side of the equation. 2.142857143 + 3.214285714m + -2.142857143 + m2 = 0 + -2.142857143 Reorder the terms: 2.142857143 + -2.142857143 + 3.214285714m + m2 = 0 + -2.142857143 Combine like terms: 2.142857143 + -2.142857143 = 0.000000000 0.000000000 + 3.214285714m + m2 = 0 + -2.142857143 3.214285714m + m2 = 0 + -2.142857143 Combine like terms: 0 + -2.142857143 = -2.142857143 3.214285714m + m2 = -2.142857143 The m term is 3.214285714m. Take half its coefficient (1.607142857). Square it (2.582908163) and add it to both sides. Add '2.582908163' to each side of the equation. 3.214285714m + 2.582908163 + m2 = -2.142857143 + 2.582908163 Reorder the terms: 2.582908163 + 3.214285714m + m2 = -2.142857143 + 2.582908163 Combine like terms: -2.142857143 + 2.582908163 = 0.44005102 2.582908163 + 3.214285714m + m2 = 0.44005102 Factor a perfect square on the left side: (m + 1.607142857)(m + 1.607142857) = 0.44005102 Calculate the square root of the right side: 0.663363415 Break this problem into two subproblems by setting (m + 1.607142857) equal to 0.663363415 and -0.663363415.Subproblem 1
m + 1.607142857 = 0.663363415 Simplifying m + 1.607142857 = 0.663363415 Reorder the terms: 1.607142857 + m = 0.663363415 Solving 1.607142857 + m = 0.663363415 Solving for variable 'm'. Move all terms containing m to the left, all other terms to the right. Add '-1.607142857' to each side of the equation. 1.607142857 + -1.607142857 + m = 0.663363415 + -1.607142857 Combine like terms: 1.607142857 + -1.607142857 = 0.000000000 0.000000000 + m = 0.663363415 + -1.607142857 m = 0.663363415 + -1.607142857 Combine like terms: 0.663363415 + -1.607142857 = -0.943779442 m = -0.943779442 Simplifying m = -0.943779442Subproblem 2
m + 1.607142857 = -0.663363415 Simplifying m + 1.607142857 = -0.663363415 Reorder the terms: 1.607142857 + m = -0.663363415 Solving 1.607142857 + m = -0.663363415 Solving for variable 'm'. Move all terms containing m to the left, all other terms to the right. Add '-1.607142857' to each side of the equation. 1.607142857 + -1.607142857 + m = -0.663363415 + -1.607142857 Combine like terms: 1.607142857 + -1.607142857 = 0.000000000 0.000000000 + m = -0.663363415 + -1.607142857 m = -0.663363415 + -1.607142857 Combine like terms: -0.663363415 + -1.607142857 = -2.270506272 m = -2.270506272 Simplifying m = -2.270506272Solution
The solution to the problem is based on the solutions from the subproblems. m = {-0.943779442, -2.270506272}
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