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Simplifying 1 = (0.05 + -1x)(0.05 + -1x) Multiply (0.05 + -1x) * (0.05 + -1x) 1 = (0.05(0.05 + -1x) + -1x * (0.05 + -1x)) 1 = ((0.05 * 0.05 + -1x * 0.05) + -1x * (0.05 + -1x)) 1 = ((0.0025 + -0.05x) + -1x * (0.05 + -1x)) 1 = (0.0025 + -0.05x + (0.05 * -1x + -1x * -1x)) 1 = (0.0025 + -0.05x + (-0.05x + 1x2)) Combine like terms: -0.05x + -0.05x = -0.1x 1 = (0.0025 + -0.1x + 1x2) Solving 1 = 0.0025 + -0.1x + 1x2 Solving for variable 'x'. Combine like terms: 1 + -0.0025 = 0.9975 0.9975 + 0.1x + -1x2 = 0.0025 + -0.1x + 1x2 + -0.0025 + 0.1x + -1x2 Reorder the terms: 0.9975 + 0.1x + -1x2 = 0.0025 + -0.0025 + -0.1x + 0.1x + 1x2 + -1x2 Combine like terms: 0.0025 + -0.0025 = 0.0000 0.9975 + 0.1x + -1x2 = 0.0000 + -0.1x + 0.1x + 1x2 + -1x2 0.9975 + 0.1x + -1x2 = -0.1x + 0.1x + 1x2 + -1x2 Combine like terms: -0.1x + 0.1x = 0.0 0.9975 + 0.1x + -1x2 = 0.0 + 1x2 + -1x2 0.9975 + 0.1x + -1x2 = 1x2 + -1x2 Combine like terms: 1x2 + -1x2 = 0 0.9975 + 0.1x + -1x2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -0.9975 + -0.1x + x2 = 0 Move the constant term to the right: Add '0.9975' to each side of the equation. -0.9975 + -0.1x + 0.9975 + x2 = 0 + 0.9975 Reorder the terms: -0.9975 + 0.9975 + -0.1x + x2 = 0 + 0.9975 Combine like terms: -0.9975 + 0.9975 = 0.0000 0.0000 + -0.1x + x2 = 0 + 0.9975 -0.1x + x2 = 0 + 0.9975 Combine like terms: 0 + 0.9975 = 0.9975 -0.1x + x2 = 0.9975 The x term is -0.1x. Take half its coefficient (-0.05). Square it (0.0025) and add it to both sides. Add '0.0025' to each side of the equation. -0.1x + 0.0025 + x2 = 0.9975 + 0.0025 Reorder the terms: 0.0025 + -0.1x + x2 = 0.9975 + 0.0025 Combine like terms: 0.9975 + 0.0025 = 1 0.0025 + -0.1x + x2 = 1 Factor a perfect square on the left side: (x + -0.05)(x + -0.05) = 1 Calculate the square root of the right side: 1 Break this problem into two subproblems by setting (x + -0.05) equal to 1 and -1.Subproblem 1
x + -0.05 = 1 Simplifying x + -0.05 = 1 Reorder the terms: -0.05 + x = 1 Solving -0.05 + x = 1 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.05' to each side of the equation. -0.05 + 0.05 + x = 1 + 0.05 Combine like terms: -0.05 + 0.05 = 0.00 0.00 + x = 1 + 0.05 x = 1 + 0.05 Combine like terms: 1 + 0.05 = 1.05 x = 1.05 Simplifying x = 1.05Subproblem 2
x + -0.05 = -1 Simplifying x + -0.05 = -1 Reorder the terms: -0.05 + x = -1 Solving -0.05 + x = -1 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.05' to each side of the equation. -0.05 + 0.05 + x = -1 + 0.05 Combine like terms: -0.05 + 0.05 = 0.00 0.00 + x = -1 + 0.05 x = -1 + 0.05 Combine like terms: -1 + 0.05 = -0.95 x = -0.95 Simplifying x = -0.95Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.05, -0.95}
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