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Simplifying 0.1952 = (209269 + -1x) * x Reorder the terms for easier multiplication: 0.1952 = x(209269 + -1x) 0.1952 = (209269 * x + -1x * x) 0.1952 = (209269x + -1x2) Solving 0.1952 = 209269x + -1x2 Solving for variable 'x'. Reorder the terms: 0.1952 + -209269x + x2 = 209269x + -209269x + -1x2 + x2 Combine like terms: 209269x + -209269x = 0 0.1952 + -209269x + x2 = 0 + -1x2 + x2 0.1952 + -209269x + x2 = -1x2 + x2 Combine like terms: -1x2 + x2 = 0 0.1952 + -209269x + x2 = 0 Begin completing the square. Move the constant term to the right: Add '-0.1952' to each side of the equation. 0.1952 + -209269x + -0.1952 + x2 = 0 + -0.1952 Reorder the terms: 0.1952 + -0.1952 + -209269x + x2 = 0 + -0.1952 Combine like terms: 0.1952 + -0.1952 = 0.0000 0.0000 + -209269x + x2 = 0 + -0.1952 -209269x + x2 = 0 + -0.1952 Combine like terms: 0 + -0.1952 = -0.1952 -209269x + x2 = -0.1952 The x term is -209269x. Take half its coefficient (-104634.5). Square it (10948378590) and add it to both sides. Add '10948378590' to each side of the equation. -209269x + 10948378590 + x2 = -0.1952 + 10948378590 Reorder the terms: 10948378590 + -209269x + x2 = -0.1952 + 10948378590 Combine like terms: -0.1952 + 10948378590 = 10948378589.8048 10948378590 + -209269x + x2 = 10948378589.8048 Factor a perfect square on the left side: (x + -104634.5)(x + -104634.5) = 10948378589.8048 Calculate the square root of the right side: 104634.499997873 Break this problem into two subproblems by setting (x + -104634.5) equal to 104634.499997873 and -104634.499997873.Subproblem 1
x + -104634.5 = 104634.499997873 Simplifying x + -104634.5 = 104634.499997873 Reorder the terms: -104634.5 + x = 104634.499997873 Solving -104634.5 + x = 104634.499997873 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '104634.5' to each side of the equation. -104634.5 + 104634.5 + x = 104634.499997873 + 104634.5 Combine like terms: -104634.5 + 104634.5 = 0.0 0.0 + x = 104634.499997873 + 104634.5 x = 104634.499997873 + 104634.5 Combine like terms: 104634.499997873 + 104634.5 = 209268.999997873 x = 209268.999997873 Simplifying x = 209268.999997873Subproblem 2
x + -104634.5 = -104634.499997873 Simplifying x + -104634.5 = -104634.499997873 Reorder the terms: -104634.5 + x = -104634.499997873 Solving -104634.5 + x = -104634.499997873 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '104634.5' to each side of the equation. -104634.5 + 104634.5 + x = -104634.499997873 + 104634.5 Combine like terms: -104634.5 + 104634.5 = 0.0 0.0 + x = -104634.499997873 + 104634.5 x = -104634.499997873 + 104634.5 Combine like terms: -104634.499997873 + 104634.5 = 0.000002127 x = 0.000002127 Simplifying x = 0.000002127Solution
The solution to the problem is based on the solutions from the subproblems. x = {209268.999997873, 0.000002127}
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