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Simplifying 128x2 + -106x + -60 = 0 Reorder the terms: -60 + -106x + 128x2 = 0 Solving -60 + -106x + 128x2 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), '2'. 2(-30 + -53x + 64x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-30 + -53x + 64x2)' equal to zero and attempt to solve: Simplifying -30 + -53x + 64x2 = 0 Solving -30 + -53x + 64x2 = 0 Begin completing the square. Divide all terms by 64 the coefficient of the squared term: Divide each side by '64'. -0.46875 + -0.828125x + x2 = 0 Move the constant term to the right: Add '0.46875' to each side of the equation. -0.46875 + -0.828125x + 0.46875 + x2 = 0 + 0.46875 Reorder the terms: -0.46875 + 0.46875 + -0.828125x + x2 = 0 + 0.46875 Combine like terms: -0.46875 + 0.46875 = 0.00000 0.00000 + -0.828125x + x2 = 0 + 0.46875 -0.828125x + x2 = 0 + 0.46875 Combine like terms: 0 + 0.46875 = 0.46875 -0.828125x + x2 = 0.46875 The x term is -0.828125x. Take half its coefficient (-0.4140625). Square it (0.1714477539) and add it to both sides. Add '0.1714477539' to each side of the equation. -0.828125x + 0.1714477539 + x2 = 0.46875 + 0.1714477539 Reorder the terms: 0.1714477539 + -0.828125x + x2 = 0.46875 + 0.1714477539 Combine like terms: 0.46875 + 0.1714477539 = 0.6401977539 0.1714477539 + -0.828125x + x2 = 0.6401977539 Factor a perfect square on the left side: (x + -0.4140625)(x + -0.4140625) = 0.6401977539 Calculate the square root of the right side: 0.800123587 Break this problem into two subproblems by setting (x + -0.4140625) equal to 0.800123587 and -0.800123587.Subproblem 1
x + -0.4140625 = 0.800123587 Simplifying x + -0.4140625 = 0.800123587 Reorder the terms: -0.4140625 + x = 0.800123587 Solving -0.4140625 + x = 0.800123587 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.4140625' to each side of the equation. -0.4140625 + 0.4140625 + x = 0.800123587 + 0.4140625 Combine like terms: -0.4140625 + 0.4140625 = 0.0000000 0.0000000 + x = 0.800123587 + 0.4140625 x = 0.800123587 + 0.4140625 Combine like terms: 0.800123587 + 0.4140625 = 1.214186087 x = 1.214186087 Simplifying x = 1.214186087Subproblem 2
x + -0.4140625 = -0.800123587 Simplifying x + -0.4140625 = -0.800123587 Reorder the terms: -0.4140625 + x = -0.800123587 Solving -0.4140625 + x = -0.800123587 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.4140625' to each side of the equation. -0.4140625 + 0.4140625 + x = -0.800123587 + 0.4140625 Combine like terms: -0.4140625 + 0.4140625 = 0.0000000 0.0000000 + x = -0.800123587 + 0.4140625 x = -0.800123587 + 0.4140625 Combine like terms: -0.800123587 + 0.4140625 = -0.386061087 x = -0.386061087 Simplifying x = -0.386061087Solution
The solution to the problem is based on the solutions from the subproblems. x = {1.214186087, -0.386061087}Solution
x = {1.214186087, -0.386061087}
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