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Simplifying 32x2 + 27x + 1 = 0 Reorder the terms: 1 + 27x + 32x2 = 0 Solving 1 + 27x + 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'. 0.03125 + 0.84375x + x2 = 0 Move the constant term to the right: Add '-0.03125' to each side of the equation. 0.03125 + 0.84375x + -0.03125 + x2 = 0 + -0.03125 Reorder the terms: 0.03125 + -0.03125 + 0.84375x + x2 = 0 + -0.03125 Combine like terms: 0.03125 + -0.03125 = 0.00000 0.00000 + 0.84375x + x2 = 0 + -0.03125 0.84375x + x2 = 0 + -0.03125 Combine like terms: 0 + -0.03125 = -0.03125 0.84375x + x2 = -0.03125 The x term is 0.84375x. Take half its coefficient (0.421875). Square it (0.1779785156) and add it to both sides. Add '0.1779785156' to each side of the equation. 0.84375x + 0.1779785156 + x2 = -0.03125 + 0.1779785156 Reorder the terms: 0.1779785156 + 0.84375x + x2 = -0.03125 + 0.1779785156 Combine like terms: -0.03125 + 0.1779785156 = 0.1467285156 0.1779785156 + 0.84375x + x2 = 0.1467285156 Factor a perfect square on the left side: (x + 0.421875)(x + 0.421875) = 0.1467285156 Calculate the square root of the right side: 0.383051583 Break this problem into two subproblems by setting (x + 0.421875) equal to 0.383051583 and -0.383051583.Subproblem 1
x + 0.421875 = 0.383051583 Simplifying x + 0.421875 = 0.383051583 Reorder the terms: 0.421875 + x = 0.383051583 Solving 0.421875 + x = 0.383051583 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.421875' to each side of the equation. 0.421875 + -0.421875 + x = 0.383051583 + -0.421875 Combine like terms: 0.421875 + -0.421875 = 0.000000 0.000000 + x = 0.383051583 + -0.421875 x = 0.383051583 + -0.421875 Combine like terms: 0.383051583 + -0.421875 = -0.038823417 x = -0.038823417 Simplifying x = -0.038823417Subproblem 2
x + 0.421875 = -0.383051583 Simplifying x + 0.421875 = -0.383051583 Reorder the terms: 0.421875 + x = -0.383051583 Solving 0.421875 + x = -0.383051583 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.421875' to each side of the equation. 0.421875 + -0.421875 + x = -0.383051583 + -0.421875 Combine like terms: 0.421875 + -0.421875 = 0.000000 0.000000 + x = -0.383051583 + -0.421875 x = -0.383051583 + -0.421875 Combine like terms: -0.383051583 + -0.421875 = -0.804926583 x = -0.804926583 Simplifying x = -0.804926583Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.038823417, -0.804926583}
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