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Simplifying 1 + -1(1 + -1x)(1 + -1x) = x + 0.2 Multiply (1 + -1x) * (1 + -1x) 1 + -1(1(1 + -1x) + -1x * (1 + -1x)) = x + 0.2 1 + -1((1 * 1 + -1x * 1) + -1x * (1 + -1x)) = x + 0.2 1 + -1((1 + -1x) + -1x * (1 + -1x)) = x + 0.2 1 + -1(1 + -1x + (1 * -1x + -1x * -1x)) = x + 0.2 1 + -1(1 + -1x + (-1x + 1x2)) = x + 0.2 Combine like terms: -1x + -1x = -2x 1 + -1(1 + -2x + 1x2) = x + 0.2 1 + (1 * -1 + -2x * -1 + 1x2 * -1) = x + 0.2 1 + (-1 + 2x + -1x2) = x + 0.2 Combine like terms: 1 + -1 = 0 0 + 2x + -1x2 = x + 0.2 2x + -1x2 = x + 0.2 Reorder the terms: 2x + -1x2 = 0.2 + x Solving 2x + -1x2 = 0.2 + x Solving for variable 'x'. Reorder the terms: -0.2 + 2x + -1x + -1x2 = 0.2 + x + -0.2 + -1x Combine like terms: 2x + -1x = 1x -0.2 + 1x + -1x2 = 0.2 + x + -0.2 + -1x Reorder the terms: -0.2 + 1x + -1x2 = 0.2 + -0.2 + x + -1x Combine like terms: 0.2 + -0.2 = 0.0 -0.2 + 1x + -1x2 = 0.0 + x + -1x -0.2 + 1x + -1x2 = x + -1x Combine like terms: x + -1x = 0 -0.2 + 1x + -1x2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. 0.2 + -1x + x2 = 0 Move the constant term to the right: Add '-0.2' to each side of the equation. 0.2 + -1x + -0.2 + x2 = 0 + -0.2 Reorder the terms: 0.2 + -0.2 + -1x + x2 = 0 + -0.2 Combine like terms: 0.2 + -0.2 = 0.0 0.0 + -1x + x2 = 0 + -0.2 -1x + x2 = 0 + -0.2 Combine like terms: 0 + -0.2 = -0.2 -1x + x2 = -0.2 The x term is -1x. Take half its coefficient (-0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. -1x + 0.25 + x2 = -0.2 + 0.25 Reorder the terms: 0.25 + -1x + x2 = -0.2 + 0.25 Combine like terms: -0.2 + 0.25 = 0.05 0.25 + -1x + x2 = 0.05 Factor a perfect square on the left side: (x + -0.5)(x + -0.5) = 0.05 Calculate the square root of the right side: 0.223606798 Break this problem into two subproblems by setting (x + -0.5) equal to 0.223606798 and -0.223606798.Subproblem 1
x + -0.5 = 0.223606798 Simplifying x + -0.5 = 0.223606798 Reorder the terms: -0.5 + x = 0.223606798 Solving -0.5 + x = 0.223606798 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + x = 0.223606798 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = 0.223606798 + 0.5 x = 0.223606798 + 0.5 Combine like terms: 0.223606798 + 0.5 = 0.723606798 x = 0.723606798 Simplifying x = 0.723606798Subproblem 2
x + -0.5 = -0.223606798 Simplifying x + -0.5 = -0.223606798 Reorder the terms: -0.5 + x = -0.223606798 Solving -0.5 + x = -0.223606798 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + x = -0.223606798 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = -0.223606798 + 0.5 x = -0.223606798 + 0.5 Combine like terms: -0.223606798 + 0.5 = 0.276393202 x = 0.276393202 Simplifying x = 0.276393202Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.723606798, 0.276393202}
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