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Simplifying 0 = 14x2 + -14x + 1 Reorder the terms: 0 = 1 + -14x + 14x2 Solving 0 = 1 + -14x + 14x2 Solving for variable 'x'. Combine like terms: 0 + -1 = -1 -1 + 14x + -14x2 = 1 + -14x + 14x2 + -1 + 14x + -14x2 Reorder the terms: -1 + 14x + -14x2 = 1 + -1 + -14x + 14x + 14x2 + -14x2 Combine like terms: 1 + -1 = 0 -1 + 14x + -14x2 = 0 + -14x + 14x + 14x2 + -14x2 -1 + 14x + -14x2 = -14x + 14x + 14x2 + -14x2 Combine like terms: -14x + 14x = 0 -1 + 14x + -14x2 = 0 + 14x2 + -14x2 -1 + 14x + -14x2 = 14x2 + -14x2 Combine like terms: 14x2 + -14x2 = 0 -1 + 14x + -14x2 = 0 Begin completing the square. Divide all terms by -14 the coefficient of the squared term: Divide each side by '-14'. 0.07142857143 + -1x + x2 = 0 Move the constant term to the right: Add '-0.07142857143' to each side of the equation. 0.07142857143 + -1x + -0.07142857143 + x2 = 0 + -0.07142857143 Reorder the terms: 0.07142857143 + -0.07142857143 + -1x + x2 = 0 + -0.07142857143 Combine like terms: 0.07142857143 + -0.07142857143 = 0.00000000000 0.00000000000 + -1x + x2 = 0 + -0.07142857143 -1x + x2 = 0 + -0.07142857143 Combine like terms: 0 + -0.07142857143 = -0.07142857143 -1x + x2 = -0.07142857143 The x term is -1x. 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. -1x + 0.25 + x2 = -0.07142857143 + 0.25 Reorder the terms: 0.25 + -1x + x2 = -0.07142857143 + 0.25 Combine like terms: -0.07142857143 + 0.25 = 0.17857142857 0.25 + -1x + x2 = 0.17857142857 Factor a perfect square on the left side: (x + -0.5)(x + -0.5) = 0.17857142857 Calculate the square root of the right side: 0.422577127 Break this problem into two subproblems by setting (x + -0.5) equal to 0.422577127 and -0.422577127.Subproblem 1
x + -0.5 = 0.422577127 Simplifying x + -0.5 = 0.422577127 Reorder the terms: -0.5 + x = 0.422577127 Solving -0.5 + x = 0.422577127 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 = 0.422577127 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = 0.422577127 + 0.5 x = 0.422577127 + 0.5 Combine like terms: 0.422577127 + 0.5 = 0.922577127 x = 0.922577127 Simplifying x = 0.922577127Subproblem 2
x + -0.5 = -0.422577127 Simplifying x + -0.5 = -0.422577127 Reorder the terms: -0.5 + x = -0.422577127 Solving -0.5 + x = -0.422577127 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 = -0.422577127 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + x = -0.422577127 + 0.5 x = -0.422577127 + 0.5 Combine like terms: -0.422577127 + 0.5 = 0.077422873 x = 0.077422873 Simplifying x = 0.077422873Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.922577127, 0.077422873}
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