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Simplifying 0.25x + 0.50x(67 + -1x) = 23.00 0.25x + (67 * 0.50x + -1x * 0.50x) = 23.00 0.25x + (33.5x + -0.5x2) = 23.00 Combine like terms: 0.25x + 33.5x = 33.75x 33.75x + -0.5x2 = 23.00 Solving 33.75x + -0.5x2 = 23.00 Solving for variable 'x'. Reorder the terms: -23.00 + 33.75x + -0.5x2 = 23.00 + -23.00 Combine like terms: 23.00 + -23.00 = 0.00 -23.00 + 33.75x + -0.5x2 = 0.00 Begin completing the square. Divide all terms by -0.5 the coefficient of the squared term: Divide each side by '-0.5'. 46 + -67.5x + x2 = 0.0 Move the constant term to the right: Add '-46' to each side of the equation. 46 + -67.5x + -46 + x2 = 0.0 + -46 Reorder the terms: 46 + -46 + -67.5x + x2 = 0.0 + -46 Combine like terms: 46 + -46 = 0 0 + -67.5x + x2 = 0.0 + -46 -67.5x + x2 = 0.0 + -46 Combine like terms: 0.0 + -46 = -46 -67.5x + x2 = -46 The x term is -67.5x. Take half its coefficient (-33.75). Square it (1139.0625) and add it to both sides. Add '1139.0625' to each side of the equation. -67.5x + 1139.0625 + x2 = -46 + 1139.0625 Reorder the terms: 1139.0625 + -67.5x + x2 = -46 + 1139.0625 Combine like terms: -46 + 1139.0625 = 1093.0625 1139.0625 + -67.5x + x2 = 1093.0625 Factor a perfect square on the left side: (x + -33.75)(x + -33.75) = 1093.0625 Calculate the square root of the right side: 33.061495731 Break this problem into two subproblems by setting (x + -33.75) equal to 33.061495731 and -33.061495731.Subproblem 1
x + -33.75 = 33.061495731 Simplifying x + -33.75 = 33.061495731 Reorder the terms: -33.75 + x = 33.061495731 Solving -33.75 + x = 33.061495731 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '33.75' to each side of the equation. -33.75 + 33.75 + x = 33.061495731 + 33.75 Combine like terms: -33.75 + 33.75 = 0.00 0.00 + x = 33.061495731 + 33.75 x = 33.061495731 + 33.75 Combine like terms: 33.061495731 + 33.75 = 66.811495731 x = 66.811495731 Simplifying x = 66.811495731Subproblem 2
x + -33.75 = -33.061495731 Simplifying x + -33.75 = -33.061495731 Reorder the terms: -33.75 + x = -33.061495731 Solving -33.75 + x = -33.061495731 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '33.75' to each side of the equation. -33.75 + 33.75 + x = -33.061495731 + 33.75 Combine like terms: -33.75 + 33.75 = 0.00 0.00 + x = -33.061495731 + 33.75 x = -33.061495731 + 33.75 Combine like terms: -33.061495731 + 33.75 = 0.688504269 x = 0.688504269 Simplifying x = 0.688504269Solution
The solution to the problem is based on the solutions from the subproblems. x = {66.811495731, 0.688504269}
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