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Simplifying 14x2 + -40x = 3 Reorder the terms: -40x + 14x2 = 3 Solving -40x + 14x2 = 3 Solving for variable 'x'. Reorder the terms: -3 + -40x + 14x2 = 3 + -3 Combine like terms: 3 + -3 = 0 -3 + -40x + 14x2 = 0 Begin completing the square. Divide all terms by 14 the coefficient of the squared term: Divide each side by '14'. -0.2142857143 + -2.857142857x + x2 = 0 Move the constant term to the right: Add '0.2142857143' to each side of the equation. -0.2142857143 + -2.857142857x + 0.2142857143 + x2 = 0 + 0.2142857143 Reorder the terms: -0.2142857143 + 0.2142857143 + -2.857142857x + x2 = 0 + 0.2142857143 Combine like terms: -0.2142857143 + 0.2142857143 = 0.0000000000 0.0000000000 + -2.857142857x + x2 = 0 + 0.2142857143 -2.857142857x + x2 = 0 + 0.2142857143 Combine like terms: 0 + 0.2142857143 = 0.2142857143 -2.857142857x + x2 = 0.2142857143 The x term is -2.857142857x. Take half its coefficient (-1.428571429). Square it (2.040816328) and add it to both sides. Add '2.040816328' to each side of the equation. -2.857142857x + 2.040816328 + x2 = 0.2142857143 + 2.040816328 Reorder the terms: 2.040816328 + -2.857142857x + x2 = 0.2142857143 + 2.040816328 Combine like terms: 0.2142857143 + 2.040816328 = 2.2551020423 2.040816328 + -2.857142857x + x2 = 2.2551020423 Factor a perfect square on the left side: (x + -1.428571429)(x + -1.428571429) = 2.2551020423 Calculate the square root of the right side: 1.501699718 Break this problem into two subproblems by setting (x + -1.428571429) equal to 1.501699718 and -1.501699718.Subproblem 1
x + -1.428571429 = 1.501699718 Simplifying x + -1.428571429 = 1.501699718 Reorder the terms: -1.428571429 + x = 1.501699718 Solving -1.428571429 + x = 1.501699718 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.428571429' to each side of the equation. -1.428571429 + 1.428571429 + x = 1.501699718 + 1.428571429 Combine like terms: -1.428571429 + 1.428571429 = 0.000000000 0.000000000 + x = 1.501699718 + 1.428571429 x = 1.501699718 + 1.428571429 Combine like terms: 1.501699718 + 1.428571429 = 2.930271147 x = 2.930271147 Simplifying x = 2.930271147Subproblem 2
x + -1.428571429 = -1.501699718 Simplifying x + -1.428571429 = -1.501699718 Reorder the terms: -1.428571429 + x = -1.501699718 Solving -1.428571429 + x = -1.501699718 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '1.428571429' to each side of the equation. -1.428571429 + 1.428571429 + x = -1.501699718 + 1.428571429 Combine like terms: -1.428571429 + 1.428571429 = 0.000000000 0.000000000 + x = -1.501699718 + 1.428571429 x = -1.501699718 + 1.428571429 Combine like terms: -1.501699718 + 1.428571429 = -0.073128289 x = -0.073128289 Simplifying x = -0.073128289Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.930271147, -0.073128289}
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