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Simplifying 14x2 + 16x + -27 = 0 Reorder the terms: -27 + 16x + 14x2 = 0 Solving -27 + 16x + 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 + 1.142857143x + x2 = 0 Move the constant term to the right: Add '1.928571429' to each side of the equation. -1.928571429 + 1.142857143x + 1.928571429 + x2 = 0 + 1.928571429 Reorder the terms: -1.928571429 + 1.928571429 + 1.142857143x + x2 = 0 + 1.928571429 Combine like terms: -1.928571429 + 1.928571429 = 0.000000000 0.000000000 + 1.142857143x + x2 = 0 + 1.928571429 1.142857143x + x2 = 0 + 1.928571429 Combine like terms: 0 + 1.928571429 = 1.928571429 1.142857143x + x2 = 1.928571429 The x term is 1.142857143x. Take half its coefficient (0.5714285715). Square it (0.3265306123) and add it to both sides. Add '0.3265306123' to each side of the equation. 1.142857143x + 0.3265306123 + x2 = 1.928571429 + 0.3265306123 Reorder the terms: 0.3265306123 + 1.142857143x + x2 = 1.928571429 + 0.3265306123 Combine like terms: 1.928571429 + 0.3265306123 = 2.2551020413 0.3265306123 + 1.142857143x + x2 = 2.2551020413 Factor a perfect square on the left side: (x + 0.5714285715)(x + 0.5714285715) = 2.2551020413 Calculate the square root of the right side: 1.501699717 Break this problem into two subproblems by setting (x + 0.5714285715) equal to 1.501699717 and -1.501699717.Subproblem 1
x + 0.5714285715 = 1.501699717 Simplifying x + 0.5714285715 = 1.501699717 Reorder the terms: 0.5714285715 + x = 1.501699717 Solving 0.5714285715 + x = 1.501699717 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5714285715' to each side of the equation. 0.5714285715 + -0.5714285715 + x = 1.501699717 + -0.5714285715 Combine like terms: 0.5714285715 + -0.5714285715 = 0.0000000000 0.0000000000 + x = 1.501699717 + -0.5714285715 x = 1.501699717 + -0.5714285715 Combine like terms: 1.501699717 + -0.5714285715 = 0.9302711455 x = 0.9302711455 Simplifying x = 0.9302711455Subproblem 2
x + 0.5714285715 = -1.501699717 Simplifying x + 0.5714285715 = -1.501699717 Reorder the terms: 0.5714285715 + x = -1.501699717 Solving 0.5714285715 + x = -1.501699717 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5714285715' to each side of the equation. 0.5714285715 + -0.5714285715 + x = -1.501699717 + -0.5714285715 Combine like terms: 0.5714285715 + -0.5714285715 = 0.0000000000 0.0000000000 + x = -1.501699717 + -0.5714285715 x = -1.501699717 + -0.5714285715 Combine like terms: -1.501699717 + -0.5714285715 = -2.0731282885 x = -2.0731282885 Simplifying x = -2.0731282885Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.9302711455, -2.0731282885}
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