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Simplifying (0.1 + 2x)(0.1 + 2x) = 0 Multiply (0.1 + 2x) * (0.1 + 2x) (0.1(0.1 + 2x) + 2x * (0.1 + 2x)) = 0 ((0.1 * 0.1 + 2x * 0.1) + 2x * (0.1 + 2x)) = 0 ((0.01 + 0.2x) + 2x * (0.1 + 2x)) = 0 (0.01 + 0.2x + (0.1 * 2x + 2x * 2x)) = 0 (0.01 + 0.2x + (0.2x + 4x2)) = 0 Combine like terms: 0.2x + 0.2x = 0.4x (0.01 + 0.4x + 4x2) = 0 Solving 0.01 + 0.4x + 4x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. 0.0025 + 0.1x + x2 = 0 Move the constant term to the right: Add '-0.0025' to each side of the equation. 0.0025 + 0.1x + -0.0025 + x2 = 0 + -0.0025 Reorder the terms: 0.0025 + -0.0025 + 0.1x + x2 = 0 + -0.0025 Combine like terms: 0.0025 + -0.0025 = 0.0000 0.0000 + 0.1x + x2 = 0 + -0.0025 0.1x + x2 = 0 + -0.0025 Combine like terms: 0 + -0.0025 = -0.0025 0.1x + x2 = -0.0025 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 = -0.0025 + 0.0025 Reorder the terms: 0.0025 + 0.1x + x2 = -0.0025 + 0.0025 Combine like terms: -0.0025 + 0.0025 = 0.0000 0.0025 + 0.1x + x2 = 0.0000 Factor a perfect square on the left side: (x + 0.05)(x + 0.05) = 0.0000 Calculate the square root of the right side: 0.0000 Break this problem into two subproblems by setting (x + 0.05) equal to 0.0000 and 0.0000.Subproblem 1
x + 0.05 = 0.0000 Simplifying x + 0.05 = 0.0000 Reorder the terms: 0.05 + x = 0.0000 Solving 0.05 + x = 0.0000 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 = 0.0000 + -0.05 Combine like terms: 0.05 + -0.05 = 0.00 0.00 + x = 0.0000 + -0.05 x = 0.0000 + -0.05 Combine like terms: 0.0000 + -0.05 = -0.05 x = -0.05 Simplifying x = -0.05Subproblem 2
x + 0.05 = 0.0000 Simplifying x + 0.05 = 0.0000 Reorder the terms: 0.05 + x = 0.0000 Solving 0.05 + x = 0.0000 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 = 0.0000 + -0.05 Combine like terms: 0.05 + -0.05 = 0.00 0.00 + x = 0.0000 + -0.05 x = 0.0000 + -0.05 Combine like terms: 0.0000 + -0.05 = -0.05 x = -0.05 Simplifying x = -0.05Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.05, -0.05}
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