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Simplifying 32x2 + -108x + -81 = 0 Reorder the terms: -81 + -108x + 32x2 = 0 Solving -81 + -108x + 32x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 32 the coefficient of the squared term: Divide each side by '32'. -2.53125 + -3.375x + x2 = 0 Move the constant term to the right: Add '2.53125' to each side of the equation. -2.53125 + -3.375x + 2.53125 + x2 = 0 + 2.53125 Reorder the terms: -2.53125 + 2.53125 + -3.375x + x2 = 0 + 2.53125 Combine like terms: -2.53125 + 2.53125 = 0.00000 0.00000 + -3.375x + x2 = 0 + 2.53125 -3.375x + x2 = 0 + 2.53125 Combine like terms: 0 + 2.53125 = 2.53125 -3.375x + x2 = 2.53125 The x term is -3.375x. Take half its coefficient (-1.6875). Square it (2.84765625) and add it to both sides. Add '2.84765625' to each side of the equation. -3.375x + 2.84765625 + x2 = 2.53125 + 2.84765625 Reorder the terms: 2.84765625 + -3.375x + x2 = 2.53125 + 2.84765625 Combine like terms: 2.53125 + 2.84765625 = 5.37890625 2.84765625 + -3.375x + x2 = 5.37890625 Factor a perfect square on the left side: (x + -1.6875)(x + -1.6875) = 5.37890625 Calculate the square root of the right side: 2.319246914 Break this problem into two subproblems by setting (x + -1.6875) equal to 2.319246914 and -2.319246914.Subproblem 1
x + -1.6875 = 2.319246914 Simplifying x + -1.6875 = 2.319246914 Reorder the terms: -1.6875 + x = 2.319246914 Solving -1.6875 + x = 2.319246914 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.6875' to each side of the equation. -1.6875 + 1.6875 + x = 2.319246914 + 1.6875 Combine like terms: -1.6875 + 1.6875 = 0.0000 0.0000 + x = 2.319246914 + 1.6875 x = 2.319246914 + 1.6875 Combine like terms: 2.319246914 + 1.6875 = 4.006746914 x = 4.006746914 Simplifying x = 4.006746914Subproblem 2
x + -1.6875 = -2.319246914 Simplifying x + -1.6875 = -2.319246914 Reorder the terms: -1.6875 + x = -2.319246914 Solving -1.6875 + x = -2.319246914 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.6875' to each side of the equation. -1.6875 + 1.6875 + x = -2.319246914 + 1.6875 Combine like terms: -1.6875 + 1.6875 = 0.0000 0.0000 + x = -2.319246914 + 1.6875 x = -2.319246914 + 1.6875 Combine like terms: -2.319246914 + 1.6875 = -0.631746914 x = -0.631746914 Simplifying x = -0.631746914Solution
The solution to the problem is based on the solutions from the subproblems. x = {4.006746914, -0.631746914}
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