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Simplifying 0.039 = x(1 + -1x) 0.039 = (1 * x + -1x * x) 0.039 = (1x + -1x2) Solving 0.039 = 1x + -1x2 Solving for variable 'x'. Reorder the terms: 0.039 + -1x + x2 = 1x + -1x + -1x2 + x2 Combine like terms: 1x + -1x = 0 0.039 + -1x + x2 = 0 + -1x2 + x2 0.039 + -1x + x2 = -1x2 + x2 Combine like terms: -1x2 + x2 = 0 0.039 + -1x + x2 = 0 Begin completing the square. Move the constant term to the right: Add '-0.039' to each side of the equation. 0.039 + -1x + -0.039 + x2 = 0 + -0.039 Reorder the terms: 0.039 + -0.039 + -1x + x2 = 0 + -0.039 Combine like terms: 0.039 + -0.039 = 0.000 0.000 + -1x + x2 = 0 + -0.039 -1x + x2 = 0 + -0.039 Combine like terms: 0 + -0.039 = -0.039 -1x + x2 = -0.039 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 = -0.039 + 0.25 Reorder the terms: 0.25 + -1x + x2 = -0.039 + 0.25 Combine like terms: -0.039 + 0.25 = 0.211 0.25 + -1x + x2 = 0.211 Factor a perfect square on the left side: (x + -0.5)(x + -0.5) = 0.211 Calculate the square root of the right side: 0.459347363 Break this problem into two subproblems by setting (x + -0.5) equal to 0.459347363 and -0.459347363.Subproblem 1
x + -0.5 = 0.459347363 Simplifying x + -0.5 = 0.459347363 Reorder the terms: -0.5 + x = 0.459347363 Solving -0.5 + x = 0.459347363 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 = 0.459347363 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = 0.459347363 + 0.5 x = 0.459347363 + 0.5 Combine like terms: 0.459347363 + 0.5 = 0.959347363 x = 0.959347363 Simplifying x = 0.959347363Subproblem 2
x + -0.5 = -0.459347363 Simplifying x + -0.5 = -0.459347363 Reorder the terms: -0.5 + x = -0.459347363 Solving -0.5 + x = -0.459347363 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 = -0.459347363 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = -0.459347363 + 0.5 x = -0.459347363 + 0.5 Combine like terms: -0.459347363 + 0.5 = 0.040652637 x = 0.040652637 Simplifying x = 0.040652637Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.959347363, 0.040652637}
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