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Simplifying 128x2 + -64 + 13 = -14(4x + -1) Reorder the terms: -64 + 13 + 128x2 = -14(4x + -1) Combine like terms: -64 + 13 = -51 -51 + 128x2 = -14(4x + -1) Reorder the terms: -51 + 128x2 = -14(-1 + 4x) -51 + 128x2 = (-1 * -14 + 4x * -14) -51 + 128x2 = (14 + -56x) Solving -51 + 128x2 = 14 + -56x Solving for variable 'x'. Reorder the terms: -51 + -14 + 56x + 128x2 = 14 + -56x + -14 + 56x Combine like terms: -51 + -14 = -65 -65 + 56x + 128x2 = 14 + -56x + -14 + 56x Reorder the terms: -65 + 56x + 128x2 = 14 + -14 + -56x + 56x Combine like terms: 14 + -14 = 0 -65 + 56x + 128x2 = 0 + -56x + 56x -65 + 56x + 128x2 = -56x + 56x Combine like terms: -56x + 56x = 0 -65 + 56x + 128x2 = 0 Begin completing the square. Divide all terms by 128 the coefficient of the squared term: Divide each side by '128'. -0.5078125 + 0.4375x + x2 = 0 Move the constant term to the right: Add '0.5078125' to each side of the equation. -0.5078125 + 0.4375x + 0.5078125 + x2 = 0 + 0.5078125 Reorder the terms: -0.5078125 + 0.5078125 + 0.4375x + x2 = 0 + 0.5078125 Combine like terms: -0.5078125 + 0.5078125 = 0.0000000 0.0000000 + 0.4375x + x2 = 0 + 0.5078125 0.4375x + x2 = 0 + 0.5078125 Combine like terms: 0 + 0.5078125 = 0.5078125 0.4375x + x2 = 0.5078125 The x term is 0.4375x. Take half its coefficient (0.21875). Square it (0.0478515625) and add it to both sides. Add '0.0478515625' to each side of the equation. 0.4375x + 0.0478515625 + x2 = 0.5078125 + 0.0478515625 Reorder the terms: 0.0478515625 + 0.4375x + x2 = 0.5078125 + 0.0478515625 Combine like terms: 0.5078125 + 0.0478515625 = 0.5556640625 0.0478515625 + 0.4375x + x2 = 0.5556640625 Factor a perfect square on the left side: (x + 0.21875)(x + 0.21875) = 0.5556640625 Calculate the square root of the right side: 0.745428778 Break this problem into two subproblems by setting (x + 0.21875) equal to 0.745428778 and -0.745428778.Subproblem 1
x + 0.21875 = 0.745428778 Simplifying x + 0.21875 = 0.745428778 Reorder the terms: 0.21875 + x = 0.745428778 Solving 0.21875 + x = 0.745428778 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.21875' to each side of the equation. 0.21875 + -0.21875 + x = 0.745428778 + -0.21875 Combine like terms: 0.21875 + -0.21875 = 0.00000 0.00000 + x = 0.745428778 + -0.21875 x = 0.745428778 + -0.21875 Combine like terms: 0.745428778 + -0.21875 = 0.526678778 x = 0.526678778 Simplifying x = 0.526678778Subproblem 2
x + 0.21875 = -0.745428778 Simplifying x + 0.21875 = -0.745428778 Reorder the terms: 0.21875 + x = -0.745428778 Solving 0.21875 + x = -0.745428778 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.21875' to each side of the equation. 0.21875 + -0.21875 + x = -0.745428778 + -0.21875 Combine like terms: 0.21875 + -0.21875 = 0.00000 0.00000 + x = -0.745428778 + -0.21875 x = -0.745428778 + -0.21875 Combine like terms: -0.745428778 + -0.21875 = -0.964178778 x = -0.964178778 Simplifying x = -0.964178778Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.526678778, -0.964178778}
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