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Simplifying 0.137x2 + 0.054x + -0.431 = 0 Reorder the terms: -0.431 + 0.054x + 0.137x2 = 0 Solving -0.431 + 0.054x + 0.137x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 0.137 the coefficient of the squared term: Divide each side by '0.137'. -3.145985401 + 0.3941605839x + x2 = 0 Move the constant term to the right: Add '3.145985401' to each side of the equation. -3.145985401 + 0.3941605839x + 3.145985401 + x2 = 0 + 3.145985401 Reorder the terms: -3.145985401 + 3.145985401 + 0.3941605839x + x2 = 0 + 3.145985401 Combine like terms: -3.145985401 + 3.145985401 = 0.000000000 0.000000000 + 0.3941605839x + x2 = 0 + 3.145985401 0.3941605839x + x2 = 0 + 3.145985401 Combine like terms: 0 + 3.145985401 = 3.145985401 0.3941605839x + x2 = 3.145985401 The x term is 0.3941605839x. Take half its coefficient (0.197080292). Square it (0.03884064149) and add it to both sides. Add '0.03884064149' to each side of the equation. 0.3941605839x + 0.03884064149 + x2 = 3.145985401 + 0.03884064149 Reorder the terms: 0.03884064149 + 0.3941605839x + x2 = 3.145985401 + 0.03884064149 Combine like terms: 3.145985401 + 0.03884064149 = 3.18482604249 0.03884064149 + 0.3941605839x + x2 = 3.18482604249 Factor a perfect square on the left side: (x + 0.197080292)(x + 0.197080292) = 3.18482604249 Calculate the square root of the right side: 1.784608092 Break this problem into two subproblems by setting (x + 0.197080292) equal to 1.784608092 and -1.784608092.Subproblem 1
x + 0.197080292 = 1.784608092 Simplifying x + 0.197080292 = 1.784608092 Reorder the terms: 0.197080292 + x = 1.784608092 Solving 0.197080292 + x = 1.784608092 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.197080292' to each side of the equation. 0.197080292 + -0.197080292 + x = 1.784608092 + -0.197080292 Combine like terms: 0.197080292 + -0.197080292 = 0.000000000 0.000000000 + x = 1.784608092 + -0.197080292 x = 1.784608092 + -0.197080292 Combine like terms: 1.784608092 + -0.197080292 = 1.5875278 x = 1.5875278 Simplifying x = 1.5875278Subproblem 2
x + 0.197080292 = -1.784608092 Simplifying x + 0.197080292 = -1.784608092 Reorder the terms: 0.197080292 + x = -1.784608092 Solving 0.197080292 + x = -1.784608092 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.197080292' to each side of the equation. 0.197080292 + -0.197080292 + x = -1.784608092 + -0.197080292 Combine like terms: 0.197080292 + -0.197080292 = 0.000000000 0.000000000 + x = -1.784608092 + -0.197080292 x = -1.784608092 + -0.197080292 Combine like terms: -1.784608092 + -0.197080292 = -1.981688384 x = -1.981688384 Simplifying x = -1.981688384Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.5875278, -1.981688384}
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