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Simplifying 117.641 = 0.469041x2 + -3.51099x Reorder the terms: 117.641 = -3.51099x + 0.469041x2 Solving 117.641 = -3.51099x + 0.469041x2 Solving for variable 'x'. Reorder the terms: 117.641 + 3.51099x + -0.469041x2 = -3.51099x + 3.51099x + 0.469041x2 + -0.469041x2 Combine like terms: -3.51099x + 3.51099x = 0.00000 117.641 + 3.51099x + -0.469041x2 = 0.00000 + 0.469041x2 + -0.469041x2 117.641 + 3.51099x + -0.469041x2 = 0.469041x2 + -0.469041x2 Combine like terms: 0.469041x2 + -0.469041x2 = 0.000000 117.641 + 3.51099x + -0.469041x2 = 0.000000 Begin completing the square. Divide all terms by -0.469041 the coefficient of the squared term: Divide each side by '-0.469041'. -250.8117627 + -7.485465023x + x2 = 0 Move the constant term to the right: Add '250.8117627' to each side of the equation. -250.8117627 + -7.485465023x + 250.8117627 + x2 = 0 + 250.8117627 Reorder the terms: -250.8117627 + 250.8117627 + -7.485465023x + x2 = 0 + 250.8117627 Combine like terms: -250.8117627 + 250.8117627 = 0.0000000 0.0000000 + -7.485465023x + x2 = 0 + 250.8117627 -7.485465023x + x2 = 0 + 250.8117627 Combine like terms: 0 + 250.8117627 = 250.8117627 -7.485465023x + x2 = 250.8117627 The x term is -7.485465023x. Take half its coefficient (-3.742732512). Square it (14.00804666) and add it to both sides. Add '14.00804666' to each side of the equation. -7.485465023x + 14.00804666 + x2 = 250.8117627 + 14.00804666 Reorder the terms: 14.00804666 + -7.485465023x + x2 = 250.8117627 + 14.00804666 Combine like terms: 250.8117627 + 14.00804666 = 264.81980936 14.00804666 + -7.485465023x + x2 = 264.81980936 Factor a perfect square on the left side: (x + -3.742732512)(x + -3.742732512) = 264.81980936 Calculate the square root of the right side: 16.273285143 Break this problem into two subproblems by setting (x + -3.742732512) equal to 16.273285143 and -16.273285143.Subproblem 1
x + -3.742732512 = 16.273285143 Simplifying x + -3.742732512 = 16.273285143 Reorder the terms: -3.742732512 + x = 16.273285143 Solving -3.742732512 + x = 16.273285143 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.742732512' to each side of the equation. -3.742732512 + 3.742732512 + x = 16.273285143 + 3.742732512 Combine like terms: -3.742732512 + 3.742732512 = 0.000000000 0.000000000 + x = 16.273285143 + 3.742732512 x = 16.273285143 + 3.742732512 Combine like terms: 16.273285143 + 3.742732512 = 20.016017655 x = 20.016017655 Simplifying x = 20.016017655Subproblem 2
x + -3.742732512 = -16.273285143 Simplifying x + -3.742732512 = -16.273285143 Reorder the terms: -3.742732512 + x = -16.273285143 Solving -3.742732512 + x = -16.273285143 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3.742732512' to each side of the equation. -3.742732512 + 3.742732512 + x = -16.273285143 + 3.742732512 Combine like terms: -3.742732512 + 3.742732512 = 0.000000000 0.000000000 + x = -16.273285143 + 3.742732512 x = -16.273285143 + 3.742732512 Combine like terms: -16.273285143 + 3.742732512 = -12.530552631 x = -12.530552631 Simplifying x = -12.530552631Solution
The solution to the problem is based on the solutions from the subproblems. x = {20.016017655, -12.530552631}
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