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Simplifying -41x2 + 180x + -164 = 0 Reorder the terms: -164 + 180x + -41x2 = 0 Solving -164 + 180x + -41x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by -41 the coefficient of the squared term: Divide each side by '-41'. 4 + -4.390243902x + x2 = 0 Move the constant term to the right: Add '-4' to each side of the equation. 4 + -4.390243902x + -4 + x2 = 0 + -4 Reorder the terms: 4 + -4 + -4.390243902x + x2 = 0 + -4 Combine like terms: 4 + -4 = 0 0 + -4.390243902x + x2 = 0 + -4 -4.390243902x + x2 = 0 + -4 Combine like terms: 0 + -4 = -4 -4.390243902x + x2 = -4 The x term is -4.390243902x. Take half its coefficient (-2.195121951). Square it (4.818560380) and add it to both sides. Add '4.818560380' to each side of the equation. -4.390243902x + 4.818560380 + x2 = -4 + 4.818560380 Reorder the terms: 4.818560380 + -4.390243902x + x2 = -4 + 4.818560380 Combine like terms: -4 + 4.818560380 = 0.81856038 4.818560380 + -4.390243902x + x2 = 0.81856038 Factor a perfect square on the left side: (x + -2.195121951)(x + -2.195121951) = 0.81856038 Calculate the square root of the right side: 0.904743267 Break this problem into two subproblems by setting (x + -2.195121951) equal to 0.904743267 and -0.904743267.Subproblem 1
x + -2.195121951 = 0.904743267 Simplifying x + -2.195121951 = 0.904743267 Reorder the terms: -2.195121951 + x = 0.904743267 Solving -2.195121951 + x = 0.904743267 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.195121951' to each side of the equation. -2.195121951 + 2.195121951 + x = 0.904743267 + 2.195121951 Combine like terms: -2.195121951 + 2.195121951 = 0.000000000 0.000000000 + x = 0.904743267 + 2.195121951 x = 0.904743267 + 2.195121951 Combine like terms: 0.904743267 + 2.195121951 = 3.099865218 x = 3.099865218 Simplifying x = 3.099865218Subproblem 2
x + -2.195121951 = -0.904743267 Simplifying x + -2.195121951 = -0.904743267 Reorder the terms: -2.195121951 + x = -0.904743267 Solving -2.195121951 + x = -0.904743267 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.195121951' to each side of the equation. -2.195121951 + 2.195121951 + x = -0.904743267 + 2.195121951 Combine like terms: -2.195121951 + 2.195121951 = 0.000000000 0.000000000 + x = -0.904743267 + 2.195121951 x = -0.904743267 + 2.195121951 Combine like terms: -0.904743267 + 2.195121951 = 1.290378684 x = 1.290378684 Simplifying x = 1.290378684Solution
The solution to the problem is based on the solutions from the subproblems. x = {3.099865218, 1.290378684}
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