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Simplifying (0.5(27 + -1x)(15)) + ((x)(15))(0.5(27 + -1x)(15)) + ((x)(15)) = 5 Reorder the terms for easier multiplication: (0.5 * 15(27 + -1x)) + ((x)(15))(0.5(27 + -1x)(15)) + ((x)(15)) = 5 Multiply 0.5 * 15 (7.5(27 + -1x)) + ((x)(15))(0.5(27 + -1x)(15)) + ((x)(15)) = 5 ((27 * 7.5 + -1x * 7.5)) + ((x)(15))(0.5(27 + -1x)(15)) + ((x)(15)) = 5 ((202.5 + -7.5x)) + ((x)(15))(0.5(27 + -1x)(15)) + ((x)(15)) = 5 (202.5 + -7.5x) + ((x)(15))(0.5(27 + -1x)(15)) + ((x)(15)) = 5 Remove parenthesis around (202.5 + -7.5x) 202.5 + -7.5x + ((x)(15))(0.5(27 + -1x)(15)) + ((x)(15)) = 5 Reorder the terms for easier multiplication: 202.5 + -7.5x + (15x)(0.5(27 + -1x)(15)) + ((x)(15)) = 5 Remove parenthesis around (15x) 202.5 + -7.5x + 15x(0.5(27 + -1x)(15)) + ((x)(15)) = 5 Reorder the terms for easier multiplication: 202.5 + -7.5x + 15x(0.5 * 15(27 + -1x)) + ((x)(15)) = 5 Multiply 0.5 * 15 202.5 + -7.5x + 15x(7.5(27 + -1x)) + ((x)(15)) = 5 202.5 + -7.5x + 15x((27 * 7.5 + -1x * 7.5)) + ((x)(15)) = 5 202.5 + -7.5x + 15x((202.5 + -7.5x)) + ((x)(15)) = 5 202.5 + -7.5x + (202.5 * 15x + -7.5x * 15x) + ((x)(15)) = 5 202.5 + -7.5x + (3037.5x + -112.5x2) + ((x)(15)) = 5 Reorder the terms for easier multiplication: 202.5 + -7.5x + 3037.5x + -112.5x2 + (15x) = 5 Reorder the terms: 202.5 + -7.5x + 3037.5x + (15x) + -112.5x2 = 5 Combine like terms: -7.5x + 3037.5x = 3030x 202.5 + 3030x + (15x) + -112.5x2 = 5 Combine like terms: 3030x + (15x) = 3045x 202.5 + 3045x + -112.5x2 = 5 Solving 202.5 + 3045x + -112.5x2 = 5 Solving for variable 'x'. Reorder the terms: 202.5 + -5 + 3045x + -112.5x2 = 5 + -5 Combine like terms: 202.5 + -5 = 197.5 197.5 + 3045x + -112.5x2 = 5 + -5 Combine like terms: 5 + -5 = 0 197.5 + 3045x + -112.5x2 = 0 Begin completing the square. Divide all terms by -112.5 the coefficient of the squared term: Divide each side by '-112.5'. -1.755555556 + -27.06666667x + x2 = 0 Move the constant term to the right: Add '1.755555556' to each side of the equation. -1.755555556 + -27.06666667x + 1.755555556 + x2 = 0 + 1.755555556 Reorder the terms: -1.755555556 + 1.755555556 + -27.06666667x + x2 = 0 + 1.755555556 Combine like terms: -1.755555556 + 1.755555556 = 0.000000000 0.000000000 + -27.06666667x + x2 = 0 + 1.755555556 -27.06666667x + x2 = 0 + 1.755555556 Combine like terms: 0 + 1.755555556 = 1.755555556 -27.06666667x + x2 = 1.755555556 The x term is -27.06666667x. Take half its coefficient (-13.53333334). Square it (183.1511113) and add it to both sides. Add '183.1511113' to each side of the equation. -27.06666667x + 183.1511113 + x2 = 1.755555556 + 183.1511113 Reorder the terms: 183.1511113 + -27.06666667x + x2 = 1.755555556 + 183.1511113 Combine like terms: 1.755555556 + 183.1511113 = 184.906666856 183.1511113 + -27.06666667x + x2 = 184.906666856 Factor a perfect square on the left side: (x + -13.53333334)(x + -13.53333334) = 184.906666856 Calculate the square root of the right side: 13.598039081 Break this problem into two subproblems by setting (x + -13.53333334) equal to 13.598039081 and -13.598039081.Subproblem 1
x + -13.53333334 = 13.598039081 Simplifying x + -13.53333334 = 13.598039081 Reorder the terms: -13.53333334 + x = 13.598039081 Solving -13.53333334 + x = 13.598039081 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '13.53333334' to each side of the equation. -13.53333334 + 13.53333334 + x = 13.598039081 + 13.53333334 Combine like terms: -13.53333334 + 13.53333334 = 0.00000000 0.00000000 + x = 13.598039081 + 13.53333334 x = 13.598039081 + 13.53333334 Combine like terms: 13.598039081 + 13.53333334 = 27.131372421 x = 27.131372421 Simplifying x = 27.131372421Subproblem 2
x + -13.53333334 = -13.598039081 Simplifying x + -13.53333334 = -13.598039081 Reorder the terms: -13.53333334 + x = -13.598039081 Solving -13.53333334 + x = -13.598039081 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '13.53333334' to each side of the equation. -13.53333334 + 13.53333334 + x = -13.598039081 + 13.53333334 Combine like terms: -13.53333334 + 13.53333334 = 0.00000000 0.00000000 + x = -13.598039081 + 13.53333334 x = -13.598039081 + 13.53333334 Combine like terms: -13.598039081 + 13.53333334 = -0.064705741 x = -0.064705741 Simplifying x = -0.064705741Solution
The solution to the problem is based on the solutions from the subproblems. x = {27.131372421, -0.064705741}
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