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Simplifying 78 = 2m(m + 3) + m Reorder the terms: 78 = 2m(3 + m) + m 78 = (3 * 2m + m * 2m) + m 78 = (6m + 2m2) + m Reorder the terms: 78 = 6m + m + 2m2 Combine like terms: 6m + m = 7m 78 = 7m + 2m2 Solving 78 = 7m + 2m2 Solving for variable 'm'. Reorder the terms: 78 + -7m + -2m2 = 7m + -7m + 2m2 + -2m2 Combine like terms: 7m + -7m = 0 78 + -7m + -2m2 = 0 + 2m2 + -2m2 78 + -7m + -2m2 = 2m2 + -2m2 Combine like terms: 2m2 + -2m2 = 0 78 + -7m + -2m2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -39 + 3.5m + m2 = 0 Move the constant term to the right: Add '39' to each side of the equation. -39 + 3.5m + 39 + m2 = 0 + 39 Reorder the terms: -39 + 39 + 3.5m + m2 = 0 + 39 Combine like terms: -39 + 39 = 0 0 + 3.5m + m2 = 0 + 39 3.5m + m2 = 0 + 39 Combine like terms: 0 + 39 = 39 3.5m + m2 = 39 The m term is 3.5m. Take half its coefficient (1.75). Square it (3.0625) and add it to both sides. Add '3.0625' to each side of the equation. 3.5m + 3.0625 + m2 = 39 + 3.0625 Reorder the terms: 3.0625 + 3.5m + m2 = 39 + 3.0625 Combine like terms: 39 + 3.0625 = 42.0625 3.0625 + 3.5m + m2 = 42.0625 Factor a perfect square on the left side: (m + 1.75)(m + 1.75) = 42.0625 Calculate the square root of the right side: 6.485560886 Break this problem into two subproblems by setting (m + 1.75) equal to 6.485560886 and -6.485560886.Subproblem 1
m + 1.75 = 6.485560886 Simplifying m + 1.75 = 6.485560886 Reorder the terms: 1.75 + m = 6.485560886 Solving 1.75 + m = 6.485560886 Solving for variable 'm'. Move all terms containing m to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + m = 6.485560886 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + m = 6.485560886 + -1.75 m = 6.485560886 + -1.75 Combine like terms: 6.485560886 + -1.75 = 4.735560886 m = 4.735560886 Simplifying m = 4.735560886Subproblem 2
m + 1.75 = -6.485560886 Simplifying m + 1.75 = -6.485560886 Reorder the terms: 1.75 + m = -6.485560886 Solving 1.75 + m = -6.485560886 Solving for variable 'm'. Move all terms containing m to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + m = -6.485560886 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + m = -6.485560886 + -1.75 m = -6.485560886 + -1.75 Combine like terms: -6.485560886 + -1.75 = -8.235560886 m = -8.235560886 Simplifying m = -8.235560886Solution
The solution to the problem is based on the solutions from the subproblems. m = {4.735560886, -8.235560886}
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