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Simplifying 0.07x + 0.09x(2x) + 0.11(3x) = 232 Remove parenthesis around (2x) 0.07x + 0.09x * 2x + 0.11(3x) = 232 Reorder the terms for easier multiplication: 0.07x + 0.09 * 2x * x + 0.11(3x) = 232 Multiply 0.09 * 2 0.07x + 0.18x * x + 0.11(3x) = 232 Multiply x * x 0.07x + 0.18x2 + 0.11(3x) = 232 Remove parenthesis around (3x) 0.07x + 0.18x2 + 0.11 * 3x = 232 Multiply 0.11 * 3 0.07x + 0.18x2 + 0.33x = 232 Reorder the terms: 0.07x + 0.33x + 0.18x2 = 232 Combine like terms: 0.07x + 0.33x = 0.4x 0.4x + 0.18x2 = 232 Solving 0.4x + 0.18x2 = 232 Solving for variable 'x'. Reorder the terms: -232 + 0.4x + 0.18x2 = 232 + -232 Combine like terms: 232 + -232 = 0 -232 + 0.4x + 0.18x2 = 0 Begin completing the square. Divide all terms by 0.18 the coefficient of the squared term: Divide each side by '0.18'. -1288.888889 + 2.222222222x + x2 = 0 Move the constant term to the right: Add '1288.888889' to each side of the equation. -1288.888889 + 2.222222222x + 1288.888889 + x2 = 0 + 1288.888889 Reorder the terms: -1288.888889 + 1288.888889 + 2.222222222x + x2 = 0 + 1288.888889 Combine like terms: -1288.888889 + 1288.888889 = 0.000000 0.000000 + 2.222222222x + x2 = 0 + 1288.888889 2.222222222x + x2 = 0 + 1288.888889 Combine like terms: 0 + 1288.888889 = 1288.888889 2.222222222x + x2 = 1288.888889 The x term is 2.222222222x. Take half its coefficient (1.111111111). Square it (1.234567901) and add it to both sides. Add '1.234567901' to each side of the equation. 2.222222222x + 1.234567901 + x2 = 1288.888889 + 1.234567901 Reorder the terms: 1.234567901 + 2.222222222x + x2 = 1288.888889 + 1.234567901 Combine like terms: 1288.888889 + 1.234567901 = 1290.123456901 1.234567901 + 2.222222222x + x2 = 1290.123456901 Factor a perfect square on the left side: (x + 1.111111111)(x + 1.111111111) = 1290.123456901 Calculate the square root of the right side: 35.918288613 Break this problem into two subproblems by setting (x + 1.111111111) equal to 35.918288613 and -35.918288613.Subproblem 1
x + 1.111111111 = 35.918288613 Simplifying x + 1.111111111 = 35.918288613 Reorder the terms: 1.111111111 + x = 35.918288613 Solving 1.111111111 + x = 35.918288613 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.111111111' to each side of the equation. 1.111111111 + -1.111111111 + x = 35.918288613 + -1.111111111 Combine like terms: 1.111111111 + -1.111111111 = 0.000000000 0.000000000 + x = 35.918288613 + -1.111111111 x = 35.918288613 + -1.111111111 Combine like terms: 35.918288613 + -1.111111111 = 34.807177502 x = 34.807177502 Simplifying x = 34.807177502Subproblem 2
x + 1.111111111 = -35.918288613 Simplifying x + 1.111111111 = -35.918288613 Reorder the terms: 1.111111111 + x = -35.918288613 Solving 1.111111111 + x = -35.918288613 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.111111111' to each side of the equation. 1.111111111 + -1.111111111 + x = -35.918288613 + -1.111111111 Combine like terms: 1.111111111 + -1.111111111 = 0.000000000 0.000000000 + x = -35.918288613 + -1.111111111 x = -35.918288613 + -1.111111111 Combine like terms: -35.918288613 + -1.111111111 = -37.029399724 x = -37.029399724 Simplifying x = -37.029399724Solution
The solution to the problem is based on the solutions from the subproblems. x = {34.807177502, -37.029399724}
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