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Simplifying 5(c + 78) = 9c(45 + -1c) + 12c Reorder the terms: 5(78 + c) = 9c(45 + -1c) + 12c (78 * 5 + c * 5) = 9c(45 + -1c) + 12c (390 + 5c) = 9c(45 + -1c) + 12c 390 + 5c = (45 * 9c + -1c * 9c) + 12c 390 + 5c = (405c + -9c2) + 12c Reorder the terms: 390 + 5c = 405c + 12c + -9c2 Combine like terms: 405c + 12c = 417c 390 + 5c = 417c + -9c2 Solving 390 + 5c = 417c + -9c2 Solving for variable 'c'. Combine like terms: 5c + -417c = -412c 390 + -412c + 9c2 = 417c + -9c2 + -417c + 9c2 Reorder the terms: 390 + -412c + 9c2 = 417c + -417c + -9c2 + 9c2 Combine like terms: 417c + -417c = 0 390 + -412c + 9c2 = 0 + -9c2 + 9c2 390 + -412c + 9c2 = -9c2 + 9c2 Combine like terms: -9c2 + 9c2 = 0 390 + -412c + 9c2 = 0 Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. 43.33333333 + -45.77777778c + c2 = 0 Move the constant term to the right: Add '-43.33333333' to each side of the equation. 43.33333333 + -45.77777778c + -43.33333333 + c2 = 0 + -43.33333333 Reorder the terms: 43.33333333 + -43.33333333 + -45.77777778c + c2 = 0 + -43.33333333 Combine like terms: 43.33333333 + -43.33333333 = 0.00000000 0.00000000 + -45.77777778c + c2 = 0 + -43.33333333 -45.77777778c + c2 = 0 + -43.33333333 Combine like terms: 0 + -43.33333333 = -43.33333333 -45.77777778c + c2 = -43.33333333 The c term is -45.77777778c. Take half its coefficient (-22.88888889). Square it (523.9012346) and add it to both sides. Add '523.9012346' to each side of the equation. -45.77777778c + 523.9012346 + c2 = -43.33333333 + 523.9012346 Reorder the terms: 523.9012346 + -45.77777778c + c2 = -43.33333333 + 523.9012346 Combine like terms: -43.33333333 + 523.9012346 = 480.56790127 523.9012346 + -45.77777778c + c2 = 480.56790127 Factor a perfect square on the left side: (c + -22.88888889)(c + -22.88888889) = 480.56790127 Calculate the square root of the right side: 21.921858983 Break this problem into two subproblems by setting (c + -22.88888889) equal to 21.921858983 and -21.921858983.Subproblem 1
c + -22.88888889 = 21.921858983 Simplifying c + -22.88888889 = 21.921858983 Reorder the terms: -22.88888889 + c = 21.921858983 Solving -22.88888889 + c = 21.921858983 Solving for variable 'c'. Move all terms containing c to the left, all other terms to the right. Add '22.88888889' to each side of the equation. -22.88888889 + 22.88888889 + c = 21.921858983 + 22.88888889 Combine like terms: -22.88888889 + 22.88888889 = 0.00000000 0.00000000 + c = 21.921858983 + 22.88888889 c = 21.921858983 + 22.88888889 Combine like terms: 21.921858983 + 22.88888889 = 44.810747873 c = 44.810747873 Simplifying c = 44.810747873Subproblem 2
c + -22.88888889 = -21.921858983 Simplifying c + -22.88888889 = -21.921858983 Reorder the terms: -22.88888889 + c = -21.921858983 Solving -22.88888889 + c = -21.921858983 Solving for variable 'c'. Move all terms containing c to the left, all other terms to the right. Add '22.88888889' to each side of the equation. -22.88888889 + 22.88888889 + c = -21.921858983 + 22.88888889 Combine like terms: -22.88888889 + 22.88888889 = 0.00000000 0.00000000 + c = -21.921858983 + 22.88888889 c = -21.921858983 + 22.88888889 Combine like terms: -21.921858983 + 22.88888889 = 0.967029907 c = 0.967029907 Simplifying c = 0.967029907Solution
The solution to the problem is based on the solutions from the subproblems. c = {44.810747873, 0.967029907}
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