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Simplifying 81x2 + -108x + -29 = 0 Reorder the terms: -29 + -108x + 81x2 = 0 Solving -29 + -108x + 81x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 81 the coefficient of the squared term: Divide each side by '81'. -0.3580246914 + -1.333333333x + x2 = 0 Move the constant term to the right: Add '0.3580246914' to each side of the equation. -0.3580246914 + -1.333333333x + 0.3580246914 + x2 = 0 + 0.3580246914 Reorder the terms: -0.3580246914 + 0.3580246914 + -1.333333333x + x2 = 0 + 0.3580246914 Combine like terms: -0.3580246914 + 0.3580246914 = 0.0000000000 0.0000000000 + -1.333333333x + x2 = 0 + 0.3580246914 -1.333333333x + x2 = 0 + 0.3580246914 Combine like terms: 0 + 0.3580246914 = 0.3580246914 -1.333333333x + x2 = 0.3580246914 The x term is -1.333333333x. Take half its coefficient (-0.6666666665). Square it (0.4444444442) and add it to both sides. Add '0.4444444442' to each side of the equation. -1.333333333x + 0.4444444442 + x2 = 0.3580246914 + 0.4444444442 Reorder the terms: 0.4444444442 + -1.333333333x + x2 = 0.3580246914 + 0.4444444442 Combine like terms: 0.3580246914 + 0.4444444442 = 0.8024691356 0.4444444442 + -1.333333333x + x2 = 0.8024691356 Factor a perfect square on the left side: (x + -0.6666666665)(x + -0.6666666665) = 0.8024691356 Calculate the square root of the right side: 0.895806416 Break this problem into two subproblems by setting (x + -0.6666666665) equal to 0.895806416 and -0.895806416.Subproblem 1
x + -0.6666666665 = 0.895806416 Simplifying x + -0.6666666665 = 0.895806416 Reorder the terms: -0.6666666665 + x = 0.895806416 Solving -0.6666666665 + x = 0.895806416 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.6666666665' to each side of the equation. -0.6666666665 + 0.6666666665 + x = 0.895806416 + 0.6666666665 Combine like terms: -0.6666666665 + 0.6666666665 = 0.0000000000 0.0000000000 + x = 0.895806416 + 0.6666666665 x = 0.895806416 + 0.6666666665 Combine like terms: 0.895806416 + 0.6666666665 = 1.5624730825 x = 1.5624730825 Simplifying x = 1.5624730825Subproblem 2
x + -0.6666666665 = -0.895806416 Simplifying x + -0.6666666665 = -0.895806416 Reorder the terms: -0.6666666665 + x = -0.895806416 Solving -0.6666666665 + x = -0.895806416 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.6666666665' to each side of the equation. -0.6666666665 + 0.6666666665 + x = -0.895806416 + 0.6666666665 Combine like terms: -0.6666666665 + 0.6666666665 = 0.0000000000 0.0000000000 + x = -0.895806416 + 0.6666666665 x = -0.895806416 + 0.6666666665 Combine like terms: -0.895806416 + 0.6666666665 = -0.2291397495 x = -0.2291397495 Simplifying x = -0.2291397495Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.5624730825, -0.2291397495}
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