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Simplifying 7.65 = 2.25 + 0.20m(m + -1) Reorder the terms: 7.65 = 2.25 + 0.20m(-1 + m) 7.65 = 2.25 + (-1 * 0.20m + m * 0.20m) 7.65 = 2.25 + (-0.2m + 0.20m2) Solving 7.65 = 2.25 + -0.2m + 0.20m2 Solving for variable 'm'. Combine like terms: 7.65 + -2.25 = 5.4 5.4 + 0.2m + -0.20m2 = 2.25 + -0.2m + 0.20m2 + -2.25 + 0.2m + -0.20m2 Reorder the terms: 5.4 + 0.2m + -0.20m2 = 2.25 + -2.25 + -0.2m + 0.2m + 0.20m2 + -0.20m2 Combine like terms: 2.25 + -2.25 = 0.00 5.4 + 0.2m + -0.20m2 = 0.00 + -0.2m + 0.2m + 0.20m2 + -0.20m2 5.4 + 0.2m + -0.20m2 = -0.2m + 0.2m + 0.20m2 + -0.20m2 Combine like terms: -0.2m + 0.2m = 0.0 5.4 + 0.2m + -0.20m2 = 0.0 + 0.20m2 + -0.20m2 5.4 + 0.2m + -0.20m2 = 0.20m2 + -0.20m2 Combine like terms: 0.20m2 + -0.20m2 = 0.00 5.4 + 0.2m + -0.20m2 = 0.00 Begin completing the square. Divide all terms by -0.20 the coefficient of the squared term: Divide each side by '-0.20'. -27 + -1m + m2 = 0 Move the constant term to the right: Add '27' to each side of the equation. -27 + -1m + 27 + m2 = 0 + 27 Reorder the terms: -27 + 27 + -1m + m2 = 0 + 27 Combine like terms: -27 + 27 = 0 0 + -1m + m2 = 0 + 27 -1m + m2 = 0 + 27 Combine like terms: 0 + 27 = 27 -1m + m2 = 27 The m term is -1m. 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 = 27 + 0.25 Reorder the terms: 0.25 + -1m + m2 = 27 + 0.25 Combine like terms: 27 + 0.25 = 27.25 0.25 + -1m + m2 = 27.25 Factor a perfect square on the left side: (m + -0.5)(m + -0.5) = 27.25 Calculate the square root of the right side: 5.220153254 Break this problem into two subproblems by setting (m + -0.5) equal to 5.220153254 and -5.220153254.Subproblem 1
m + -0.5 = 5.220153254 Simplifying m + -0.5 = 5.220153254 Reorder the terms: -0.5 + m = 5.220153254 Solving -0.5 + m = 5.220153254 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 = 5.220153254 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + m = 5.220153254 + 0.5 m = 5.220153254 + 0.5 Combine like terms: 5.220153254 + 0.5 = 5.720153254 m = 5.720153254 Simplifying m = 5.720153254Subproblem 2
m + -0.5 = -5.220153254 Simplifying m + -0.5 = -5.220153254 Reorder the terms: -0.5 + m = -5.220153254 Solving -0.5 + m = -5.220153254 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 = -5.220153254 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + m = -5.220153254 + 0.5 m = -5.220153254 + 0.5 Combine like terms: -5.220153254 + 0.5 = -4.720153254 m = -4.720153254 Simplifying m = -4.720153254Solution
The solution to the problem is based on the solutions from the subproblems. m = {5.720153254, -4.720153254}
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