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Simplifying 32x2 + 16x + 1 = 0 Reorder the terms: 1 + 16x + 32x2 = 0 Solving 1 + 16x + 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 + 0.5x + x2 = 0 Move the constant term to the right: Add '-0.03125' to each side of the equation. 0.03125 + 0.5x + -0.03125 + x2 = 0 + -0.03125 Reorder the terms: 0.03125 + -0.03125 + 0.5x + x2 = 0 + -0.03125 Combine like terms: 0.03125 + -0.03125 = 0.00000 0.00000 + 0.5x + x2 = 0 + -0.03125 0.5x + x2 = 0 + -0.03125 Combine like terms: 0 + -0.03125 = -0.03125 0.5x + x2 = -0.03125 The x term is 0.5x. Take half its coefficient (0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. 0.5x + 0.0625 + x2 = -0.03125 + 0.0625 Reorder the terms: 0.0625 + 0.5x + x2 = -0.03125 + 0.0625 Combine like terms: -0.03125 + 0.0625 = 0.03125 0.0625 + 0.5x + x2 = 0.03125 Factor a perfect square on the left side: (x + 0.25)(x + 0.25) = 0.03125 Calculate the square root of the right side: 0.176776695 Break this problem into two subproblems by setting (x + 0.25) equal to 0.176776695 and -0.176776695.Subproblem 1
x + 0.25 = 0.176776695 Simplifying x + 0.25 = 0.176776695 Reorder the terms: 0.25 + x = 0.176776695 Solving 0.25 + x = 0.176776695 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + x = 0.176776695 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = 0.176776695 + -0.25 x = 0.176776695 + -0.25 Combine like terms: 0.176776695 + -0.25 = -0.073223305 x = -0.073223305 Simplifying x = -0.073223305Subproblem 2
x + 0.25 = -0.176776695 Simplifying x + 0.25 = -0.176776695 Reorder the terms: 0.25 + x = -0.176776695 Solving 0.25 + x = -0.176776695 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + x = -0.176776695 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + x = -0.176776695 + -0.25 x = -0.176776695 + -0.25 Combine like terms: -0.176776695 + -0.25 = -0.426776695 x = -0.426776695 Simplifying x = -0.426776695Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.073223305, -0.426776695}
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