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Simplifying 0 = (-7M2 + 8M + -6) + -1(-5M2 + -5M + -10) Reorder the terms: 0 = (-6 + 8M + -7M2) + -1(-5M2 + -5M + -10) Remove parenthesis around (-6 + 8M + -7M2) 0 = -6 + 8M + -7M2 + -1(-5M2 + -5M + -10) Reorder the terms: 0 = -6 + 8M + -7M2 + -1(-10 + -5M + -5M2) 0 = -6 + 8M + -7M2 + (-10 * -1 + -5M * -1 + -5M2 * -1) 0 = -6 + 8M + -7M2 + (10 + 5M + 5M2) Reorder the terms: 0 = -6 + 10 + 8M + 5M + -7M2 + 5M2 Combine like terms: -6 + 10 = 4 0 = 4 + 8M + 5M + -7M2 + 5M2 Combine like terms: 8M + 5M = 13M 0 = 4 + 13M + -7M2 + 5M2 Combine like terms: -7M2 + 5M2 = -2M2 0 = 4 + 13M + -2M2 Solving 0 = 4 + 13M + -2M2 Solving for variable 'M'. Combine like terms: 0 + -4 = -4 -4 + -13M + 2M2 = 4 + 13M + -2M2 + -4 + -13M + 2M2 Reorder the terms: -4 + -13M + 2M2 = 4 + -4 + 13M + -13M + -2M2 + 2M2 Combine like terms: 4 + -4 = 0 -4 + -13M + 2M2 = 0 + 13M + -13M + -2M2 + 2M2 -4 + -13M + 2M2 = 13M + -13M + -2M2 + 2M2 Combine like terms: 13M + -13M = 0 -4 + -13M + 2M2 = 0 + -2M2 + 2M2 -4 + -13M + 2M2 = -2M2 + 2M2 Combine like terms: -2M2 + 2M2 = 0 -4 + -13M + 2M2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -2 + -6.5M + M2 = 0 Move the constant term to the right: Add '2' to each side of the equation. -2 + -6.5M + 2 + M2 = 0 + 2 Reorder the terms: -2 + 2 + -6.5M + M2 = 0 + 2 Combine like terms: -2 + 2 = 0 0 + -6.5M + M2 = 0 + 2 -6.5M + M2 = 0 + 2 Combine like terms: 0 + 2 = 2 -6.5M + M2 = 2 The M term is -6.5M. Take half its coefficient (-3.25). Square it (10.5625) and add it to both sides. Add '10.5625' to each side of the equation. -6.5M + 10.5625 + M2 = 2 + 10.5625 Reorder the terms: 10.5625 + -6.5M + M2 = 2 + 10.5625 Combine like terms: 2 + 10.5625 = 12.5625 10.5625 + -6.5M + M2 = 12.5625 Factor a perfect square on the left side: (M + -3.25)(M + -3.25) = 12.5625 Calculate the square root of the right side: 3.54436172 Break this problem into two subproblems by setting (M + -3.25) equal to 3.54436172 and -3.54436172.Subproblem 1
M + -3.25 = 3.54436172 Simplifying M + -3.25 = 3.54436172 Reorder the terms: -3.25 + M = 3.54436172 Solving -3.25 + M = 3.54436172 Solving for variable 'M'. Move all terms containing M to the left, all other terms to the right. Add '3.25' to each side of the equation. -3.25 + 3.25 + M = 3.54436172 + 3.25 Combine like terms: -3.25 + 3.25 = 0.00 0.00 + M = 3.54436172 + 3.25 M = 3.54436172 + 3.25 Combine like terms: 3.54436172 + 3.25 = 6.79436172 M = 6.79436172 Simplifying M = 6.79436172Subproblem 2
M + -3.25 = -3.54436172 Simplifying M + -3.25 = -3.54436172 Reorder the terms: -3.25 + M = -3.54436172 Solving -3.25 + M = -3.54436172 Solving for variable 'M'. Move all terms containing M to the left, all other terms to the right. Add '3.25' to each side of the equation. -3.25 + 3.25 + M = -3.54436172 + 3.25 Combine like terms: -3.25 + 3.25 = 0.00 0.00 + M = -3.54436172 + 3.25 M = -3.54436172 + 3.25 Combine like terms: -3.54436172 + 3.25 = -0.29436172 M = -0.29436172 Simplifying M = -0.29436172Solution
The solution to the problem is based on the solutions from the subproblems. M = {6.79436172, -0.29436172}
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