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Simplifying 14x2 + -67x + 27 = 0 Reorder the terms: 27 + -67x + 14x2 = 0 Solving 27 + -67x + 14x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 14 the coefficient of the squared term: Divide each side by '14'. 1.928571429 + -4.785714286x + x2 = 0 Move the constant term to the right: Add '-1.928571429' to each side of the equation. 1.928571429 + -4.785714286x + -1.928571429 + x2 = 0 + -1.928571429 Reorder the terms: 1.928571429 + -1.928571429 + -4.785714286x + x2 = 0 + -1.928571429 Combine like terms: 1.928571429 + -1.928571429 = 0.000000000 0.000000000 + -4.785714286x + x2 = 0 + -1.928571429 -4.785714286x + x2 = 0 + -1.928571429 Combine like terms: 0 + -1.928571429 = -1.928571429 -4.785714286x + x2 = -1.928571429 The x term is -4.785714286x. Take half its coefficient (-2.392857143). Square it (5.725765307) and add it to both sides. Add '5.725765307' to each side of the equation. -4.785714286x + 5.725765307 + x2 = -1.928571429 + 5.725765307 Reorder the terms: 5.725765307 + -4.785714286x + x2 = -1.928571429 + 5.725765307 Combine like terms: -1.928571429 + 5.725765307 = 3.797193878 5.725765307 + -4.785714286x + x2 = 3.797193878 Factor a perfect square on the left side: (x + -2.392857143)(x + -2.392857143) = 3.797193878 Calculate the square root of the right side: 1.948638981 Break this problem into two subproblems by setting (x + -2.392857143) equal to 1.948638981 and -1.948638981.Subproblem 1
x + -2.392857143 = 1.948638981 Simplifying x + -2.392857143 = 1.948638981 Reorder the terms: -2.392857143 + x = 1.948638981 Solving -2.392857143 + x = 1.948638981 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.392857143' to each side of the equation. -2.392857143 + 2.392857143 + x = 1.948638981 + 2.392857143 Combine like terms: -2.392857143 + 2.392857143 = 0.000000000 0.000000000 + x = 1.948638981 + 2.392857143 x = 1.948638981 + 2.392857143 Combine like terms: 1.948638981 + 2.392857143 = 4.341496124 x = 4.341496124 Simplifying x = 4.341496124Subproblem 2
x + -2.392857143 = -1.948638981 Simplifying x + -2.392857143 = -1.948638981 Reorder the terms: -2.392857143 + x = -1.948638981 Solving -2.392857143 + x = -1.948638981 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '2.392857143' to each side of the equation. -2.392857143 + 2.392857143 + x = -1.948638981 + 2.392857143 Combine like terms: -2.392857143 + 2.392857143 = 0.000000000 0.000000000 + x = -1.948638981 + 2.392857143 x = -1.948638981 + 2.392857143 Combine like terms: -1.948638981 + 2.392857143 = 0.444218162 x = 0.444218162 Simplifying x = 0.444218162Solution
The solution to the problem is based on the solutions from the subproblems. x = {4.341496124, 0.444218162}
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