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Simplifying 1 + -1x = -1x(2 + x) + 1.75(2 + x) 1 + -1x = (2 * -1x + x * -1x) + 1.75(2 + x) 1 + -1x = (-2x + -1x2) + 1.75(2 + x) 1 + -1x = -2x + -1x2 + (2 * 1.75 + x * 1.75) 1 + -1x = -2x + -1x2 + (3.5 + 1.75x) Reorder the terms: 1 + -1x = 3.5 + -2x + 1.75x + -1x2 Combine like terms: -2x + 1.75x = -0.25x 1 + -1x = 3.5 + -0.25x + -1x2 Solving 1 + -1x = 3.5 + -0.25x + -1x2 Solving for variable 'x'. Reorder the terms: 1 + -3.5 + -1x + 0.25x + x2 = 3.5 + -0.25x + -1x2 + -3.5 + 0.25x + x2 Combine like terms: 1 + -3.5 = -2.5 -2.5 + -1x + 0.25x + x2 = 3.5 + -0.25x + -1x2 + -3.5 + 0.25x + x2 Combine like terms: -1x + 0.25x = -0.75x -2.5 + -0.75x + x2 = 3.5 + -0.25x + -1x2 + -3.5 + 0.25x + x2 Reorder the terms: -2.5 + -0.75x + x2 = 3.5 + -3.5 + -0.25x + 0.25x + -1x2 + x2 Combine like terms: 3.5 + -3.5 = 0.0 -2.5 + -0.75x + x2 = 0.0 + -0.25x + 0.25x + -1x2 + x2 -2.5 + -0.75x + x2 = -0.25x + 0.25x + -1x2 + x2 Combine like terms: -0.25x + 0.25x = 0.00 -2.5 + -0.75x + x2 = 0.00 + -1x2 + x2 -2.5 + -0.75x + x2 = -1x2 + x2 Combine like terms: -1x2 + x2 = 0 -2.5 + -0.75x + x2 = 0 Begin completing the square. Move the constant term to the right: Add '2.5' to each side of the equation. -2.5 + -0.75x + 2.5 + x2 = 0 + 2.5 Reorder the terms: -2.5 + 2.5 + -0.75x + x2 = 0 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + -0.75x + x2 = 0 + 2.5 -0.75x + x2 = 0 + 2.5 Combine like terms: 0 + 2.5 = 2.5 -0.75x + x2 = 2.5 The x term is -0.75x. Take half its coefficient (-0.375). Square it (0.140625) and add it to both sides. Add '0.140625' to each side of the equation. -0.75x + 0.140625 + x2 = 2.5 + 0.140625 Reorder the terms: 0.140625 + -0.75x + x2 = 2.5 + 0.140625 Combine like terms: 2.5 + 0.140625 = 2.640625 0.140625 + -0.75x + x2 = 2.640625 Factor a perfect square on the left side: (x + -0.375)(x + -0.375) = 2.640625 Calculate the square root of the right side: 1.625 Break this problem into two subproblems by setting (x + -0.375) equal to 1.625 and -1.625.Subproblem 1
x + -0.375 = 1.625 Simplifying x + -0.375 = 1.625 Reorder the terms: -0.375 + x = 1.625 Solving -0.375 + x = 1.625 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.375' to each side of the equation. -0.375 + 0.375 + x = 1.625 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + x = 1.625 + 0.375 x = 1.625 + 0.375 Combine like terms: 1.625 + 0.375 = 2 x = 2 Simplifying x = 2Subproblem 2
x + -0.375 = -1.625 Simplifying x + -0.375 = -1.625 Reorder the terms: -0.375 + x = -1.625 Solving -0.375 + x = -1.625 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.375' to each side of the equation. -0.375 + 0.375 + x = -1.625 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + x = -1.625 + 0.375 x = -1.625 + 0.375 Combine like terms: -1.625 + 0.375 = -1.25 x = -1.25 Simplifying x = -1.25Solution
The solution to the problem is based on the solutions from the subproblems. x = {2, -1.25}
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