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Simplifying 0.5x2 + 1.75x = 204 Reorder the terms: 1.75x + 0.5x2 = 204 Solving 1.75x + 0.5x2 = 204 Solving for variable 'x'. Reorder the terms: -204 + 1.75x + 0.5x2 = 204 + -204 Combine like terms: 204 + -204 = 0 -204 + 1.75x + 0.5x2 = 0 Begin completing the square. Divide all terms by 0.5 the coefficient of the squared term: Divide each side by '0.5'. -408 + 3.5x + x2 = 0 Move the constant term to the right: Add '408' to each side of the equation. -408 + 3.5x + 408 + x2 = 0 + 408 Reorder the terms: -408 + 408 + 3.5x + x2 = 0 + 408 Combine like terms: -408 + 408 = 0 0 + 3.5x + x2 = 0 + 408 3.5x + x2 = 0 + 408 Combine like terms: 0 + 408 = 408 3.5x + x2 = 408 The x term is 3.5x. 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. 3.5x + 3.0625 + x2 = 408 + 3.0625 Reorder the terms: 3.0625 + 3.5x + x2 = 408 + 3.0625 Combine like terms: 408 + 3.0625 = 411.0625 3.0625 + 3.5x + x2 = 411.0625 Factor a perfect square on the left side: (x + 1.75)(x + 1.75) = 411.0625 Calculate the square root of the right side: 20.274676323 Break this problem into two subproblems by setting (x + 1.75) equal to 20.274676323 and -20.274676323.Subproblem 1
x + 1.75 = 20.274676323 Simplifying x + 1.75 = 20.274676323 Reorder the terms: 1.75 + x = 20.274676323 Solving 1.75 + x = 20.274676323 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + x = 20.274676323 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + x = 20.274676323 + -1.75 x = 20.274676323 + -1.75 Combine like terms: 20.274676323 + -1.75 = 18.524676323 x = 18.524676323 Simplifying x = 18.524676323Subproblem 2
x + 1.75 = -20.274676323 Simplifying x + 1.75 = -20.274676323 Reorder the terms: 1.75 + x = -20.274676323 Solving 1.75 + x = -20.274676323 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.75' to each side of the equation. 1.75 + -1.75 + x = -20.274676323 + -1.75 Combine like terms: 1.75 + -1.75 = 0.00 0.00 + x = -20.274676323 + -1.75 x = -20.274676323 + -1.75 Combine like terms: -20.274676323 + -1.75 = -22.024676323 x = -22.024676323 Simplifying x = -22.024676323Solution
The solution to the problem is based on the solutions from the subproblems. x = {18.524676323, -22.024676323}
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