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Simplifying 14x2 + 47x + 7 = 0 Reorder the terms: 7 + 47x + 14x2 = 0 Solving 7 + 47x + 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'. 0.5 + 3.357142857x + x2 = 0 Move the constant term to the right: Add '-0.5' to each side of the equation. 0.5 + 3.357142857x + -0.5 + x2 = 0 + -0.5 Reorder the terms: 0.5 + -0.5 + 3.357142857x + x2 = 0 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + 3.357142857x + x2 = 0 + -0.5 3.357142857x + x2 = 0 + -0.5 Combine like terms: 0 + -0.5 = -0.5 3.357142857x + x2 = -0.5 The x term is 3.357142857x. Take half its coefficient (1.678571429). Square it (2.817602042) and add it to both sides. Add '2.817602042' to each side of the equation. 3.357142857x + 2.817602042 + x2 = -0.5 + 2.817602042 Reorder the terms: 2.817602042 + 3.357142857x + x2 = -0.5 + 2.817602042 Combine like terms: -0.5 + 2.817602042 = 2.317602042 2.817602042 + 3.357142857x + x2 = 2.317602042 Factor a perfect square on the left side: (x + 1.678571429)(x + 1.678571429) = 2.317602042 Calculate the square root of the right side: 1.522367249 Break this problem into two subproblems by setting (x + 1.678571429) equal to 1.522367249 and -1.522367249.Subproblem 1
x + 1.678571429 = 1.522367249 Simplifying x + 1.678571429 = 1.522367249 Reorder the terms: 1.678571429 + x = 1.522367249 Solving 1.678571429 + x = 1.522367249 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.678571429' to each side of the equation. 1.678571429 + -1.678571429 + x = 1.522367249 + -1.678571429 Combine like terms: 1.678571429 + -1.678571429 = 0.000000000 0.000000000 + x = 1.522367249 + -1.678571429 x = 1.522367249 + -1.678571429 Combine like terms: 1.522367249 + -1.678571429 = -0.15620418 x = -0.15620418 Simplifying x = -0.15620418Subproblem 2
x + 1.678571429 = -1.522367249 Simplifying x + 1.678571429 = -1.522367249 Reorder the terms: 1.678571429 + x = -1.522367249 Solving 1.678571429 + x = -1.522367249 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.678571429' to each side of the equation. 1.678571429 + -1.678571429 + x = -1.522367249 + -1.678571429 Combine like terms: 1.678571429 + -1.678571429 = 0.000000000 0.000000000 + x = -1.522367249 + -1.678571429 x = -1.522367249 + -1.678571429 Combine like terms: -1.522367249 + -1.678571429 = -3.200938678 x = -3.200938678 Simplifying x = -3.200938678Solution
The solution to the problem is based on the solutions from the subproblems. x = {-0.15620418, -3.200938678}
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