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Simplifying 32x2 + 32x + -1 = 0 Reorder the terms: -1 + 32x + 32x2 = 0 Solving -1 + 32x + 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 + x + x2 = 0 Move the constant term to the right: Add '0.03125' to each side of the equation. -0.03125 + x + 0.03125 + x2 = 0 + 0.03125 Reorder the terms: -0.03125 + 0.03125 + x + x2 = 0 + 0.03125 Combine like terms: -0.03125 + 0.03125 = 0.00000 0.00000 + x + x2 = 0 + 0.03125 x + x2 = 0 + 0.03125 Combine like terms: 0 + 0.03125 = 0.03125 x + x2 = 0.03125 The x term is x. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. + 0.25 + x2 = 0.03125 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + x2 = 0.03125 + 0.25 Combine like terms: 0.03125 + 0.25 = 0.28125 1.25 + x2 = 0.28125 Factor a perfect square on the left side: (x + 0.5)(x + 0.5) = 0.28125 Calculate the square root of the right side: 0.530330086 Break this problem into two subproblems by setting (x + 0.5) equal to 0.530330086 and -0.530330086.Subproblem 1
x + 0.5 = 0.530330086 Simplifying x + 0.5 = 0.530330086 Reorder the terms: 0.5 + x = 0.530330086 Solving 0.5 + x = 0.530330086 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = 0.530330086 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 0.530330086 + -0.5 x = 0.530330086 + -0.5 Combine like terms: 0.530330086 + -0.5 = 0.030330086 x = 0.030330086 Simplifying x = 0.030330086Subproblem 2
x + 0.5 = -0.530330086 Simplifying x + 0.5 = -0.530330086 Reorder the terms: 0.5 + x = -0.530330086 Solving 0.5 + x = -0.530330086 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = -0.530330086 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -0.530330086 + -0.5 x = -0.530330086 + -0.5 Combine like terms: -0.530330086 + -0.5 = -1.030330086 x = -1.030330086 Simplifying x = -1.030330086Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.030330086, -1.030330086}
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