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Simplifying 14 = 0.033x2 + 1.19x + 5.2 Reorder the terms: 14 = 5.2 + 1.19x + 0.033x2 Solving 14 = 5.2 + 1.19x + 0.033x2 Solving for variable 'x'. Combine like terms: 14 + -5.2 = 8.8 8.8 + -1.19x + -0.033x2 = 5.2 + 1.19x + 0.033x2 + -5.2 + -1.19x + -0.033x2 Reorder the terms: 8.8 + -1.19x + -0.033x2 = 5.2 + -5.2 + 1.19x + -1.19x + 0.033x2 + -0.033x2 Combine like terms: 5.2 + -5.2 = 0.0 8.8 + -1.19x + -0.033x2 = 0.0 + 1.19x + -1.19x + 0.033x2 + -0.033x2 8.8 + -1.19x + -0.033x2 = 1.19x + -1.19x + 0.033x2 + -0.033x2 Combine like terms: 1.19x + -1.19x = 0.00 8.8 + -1.19x + -0.033x2 = 0.00 + 0.033x2 + -0.033x2 8.8 + -1.19x + -0.033x2 = 0.033x2 + -0.033x2 Combine like terms: 0.033x2 + -0.033x2 = 0.000 8.8 + -1.19x + -0.033x2 = 0.000 Begin completing the square. Divide all terms by -0.033 the coefficient of the squared term: Divide each side by '-0.033'. -266.6666667 + 36.06060606x + x2 = 0 Move the constant term to the right: Add '266.6666667' to each side of the equation. -266.6666667 + 36.06060606x + 266.6666667 + x2 = 0 + 266.6666667 Reorder the terms: -266.6666667 + 266.6666667 + 36.06060606x + x2 = 0 + 266.6666667 Combine like terms: -266.6666667 + 266.6666667 = 0.0000000 0.0000000 + 36.06060606x + x2 = 0 + 266.6666667 36.06060606x + x2 = 0 + 266.6666667 Combine like terms: 0 + 266.6666667 = 266.6666667 36.06060606x + x2 = 266.6666667 The x term is 36.06060606x. Take half its coefficient (18.03030303). Square it (325.0918274) and add it to both sides. Add '325.0918274' to each side of the equation. 36.06060606x + 325.0918274 + x2 = 266.6666667 + 325.0918274 Reorder the terms: 325.0918274 + 36.06060606x + x2 = 266.6666667 + 325.0918274 Combine like terms: 266.6666667 + 325.0918274 = 591.7584941 325.0918274 + 36.06060606x + x2 = 591.7584941 Factor a perfect square on the left side: (x + 18.03030303)(x + 18.03030303) = 591.7584941 Calculate the square root of the right side: 24.326086699 Break this problem into two subproblems by setting (x + 18.03030303) equal to 24.326086699 and -24.326086699.Subproblem 1
x + 18.03030303 = 24.326086699 Simplifying x + 18.03030303 = 24.326086699 Reorder the terms: 18.03030303 + x = 24.326086699 Solving 18.03030303 + x = 24.326086699 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-18.03030303' to each side of the equation. 18.03030303 + -18.03030303 + x = 24.326086699 + -18.03030303 Combine like terms: 18.03030303 + -18.03030303 = 0.00000000 0.00000000 + x = 24.326086699 + -18.03030303 x = 24.326086699 + -18.03030303 Combine like terms: 24.326086699 + -18.03030303 = 6.295783669 x = 6.295783669 Simplifying x = 6.295783669Subproblem 2
x + 18.03030303 = -24.326086699 Simplifying x + 18.03030303 = -24.326086699 Reorder the terms: 18.03030303 + x = -24.326086699 Solving 18.03030303 + x = -24.326086699 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-18.03030303' to each side of the equation. 18.03030303 + -18.03030303 + x = -24.326086699 + -18.03030303 Combine like terms: 18.03030303 + -18.03030303 = 0.00000000 0.00000000 + x = -24.326086699 + -18.03030303 x = -24.326086699 + -18.03030303 Combine like terms: -24.326086699 + -18.03030303 = -42.356389729 x = -42.356389729 Simplifying x = -42.356389729Solution
The solution to the problem is based on the solutions from the subproblems. x = {6.295783669, -42.356389729}
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