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Simplifying 32x2 + -12x + -1 = 0 Reorder the terms: -1 + -12x + 32x2 = 0 Solving -1 + -12x + 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 + -0.375x + x2 = 0 Move the constant term to the right: Add '0.03125' to each side of the equation. -0.03125 + -0.375x + 0.03125 + x2 = 0 + 0.03125 Reorder the terms: -0.03125 + 0.03125 + -0.375x + x2 = 0 + 0.03125 Combine like terms: -0.03125 + 0.03125 = 0.00000 0.00000 + -0.375x + x2 = 0 + 0.03125 -0.375x + x2 = 0 + 0.03125 Combine like terms: 0 + 0.03125 = 0.03125 -0.375x + x2 = 0.03125 The x term is -0.375x. Take half its coefficient (-0.1875). Square it (0.03515625) and add it to both sides. Add '0.03515625' to each side of the equation. -0.375x + 0.03515625 + x2 = 0.03125 + 0.03515625 Reorder the terms: 0.03515625 + -0.375x + x2 = 0.03125 + 0.03515625 Combine like terms: 0.03125 + 0.03515625 = 0.06640625 0.03515625 + -0.375x + x2 = 0.06640625 Factor a perfect square on the left side: (x + -0.1875)(x + -0.1875) = 0.06640625 Calculate the square root of the right side: 0.257694102 Break this problem into two subproblems by setting (x + -0.1875) equal to 0.257694102 and -0.257694102.Subproblem 1
x + -0.1875 = 0.257694102 Simplifying x + -0.1875 = 0.257694102 Reorder the terms: -0.1875 + x = 0.257694102 Solving -0.1875 + x = 0.257694102 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.1875' to each side of the equation. -0.1875 + 0.1875 + x = 0.257694102 + 0.1875 Combine like terms: -0.1875 + 0.1875 = 0.0000 0.0000 + x = 0.257694102 + 0.1875 x = 0.257694102 + 0.1875 Combine like terms: 0.257694102 + 0.1875 = 0.445194102 x = 0.445194102 Simplifying x = 0.445194102Subproblem 2
x + -0.1875 = -0.257694102 Simplifying x + -0.1875 = -0.257694102 Reorder the terms: -0.1875 + x = -0.257694102 Solving -0.1875 + x = -0.257694102 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.1875' to each side of the equation. -0.1875 + 0.1875 + x = -0.257694102 + 0.1875 Combine like terms: -0.1875 + 0.1875 = 0.0000 0.0000 + x = -0.257694102 + 0.1875 x = -0.257694102 + 0.1875 Combine like terms: -0.257694102 + 0.1875 = -0.070194102 x = -0.070194102 Simplifying x = -0.070194102Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.445194102, -0.070194102}
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