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Simplifying 4x2 + 27x2 + -24x + -120 = 0 Reorder the terms: -120 + -24x + 4x2 + 27x2 = 0 Combine like terms: 4x2 + 27x2 = 31x2 -120 + -24x + 31x2 = 0 Solving -120 + -24x + 31x2 = 0 Solving for variable 'x'. Begin completing the square. Divide all terms by 31 the coefficient of the squared term: Divide each side by '31'. -3.870967742 + -0.7741935484x + x2 = 0 Move the constant term to the right: Add '3.870967742' to each side of the equation. -3.870967742 + -0.7741935484x + 3.870967742 + x2 = 0 + 3.870967742 Reorder the terms: -3.870967742 + 3.870967742 + -0.7741935484x + x2 = 0 + 3.870967742 Combine like terms: -3.870967742 + 3.870967742 = 0.000000000 0.000000000 + -0.7741935484x + x2 = 0 + 3.870967742 -0.7741935484x + x2 = 0 + 3.870967742 Combine like terms: 0 + 3.870967742 = 3.870967742 -0.7741935484x + x2 = 3.870967742 The x term is -0.7741935484x. Take half its coefficient (-0.3870967742). Square it (0.1498439126) and add it to both sides. Add '0.1498439126' to each side of the equation. -0.7741935484x + 0.1498439126 + x2 = 3.870967742 + 0.1498439126 Reorder the terms: 0.1498439126 + -0.7741935484x + x2 = 3.870967742 + 0.1498439126 Combine like terms: 3.870967742 + 0.1498439126 = 4.0208116546 0.1498439126 + -0.7741935484x + x2 = 4.0208116546 Factor a perfect square on the left side: (x + -0.3870967742)(x + -0.3870967742) = 4.0208116546 Calculate the square root of the right side: 2.005196164 Break this problem into two subproblems by setting (x + -0.3870967742) equal to 2.005196164 and -2.005196164.Subproblem 1
x + -0.3870967742 = 2.005196164 Simplifying x + -0.3870967742 = 2.005196164 Reorder the terms: -0.3870967742 + x = 2.005196164 Solving -0.3870967742 + x = 2.005196164 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.3870967742' to each side of the equation. -0.3870967742 + 0.3870967742 + x = 2.005196164 + 0.3870967742 Combine like terms: -0.3870967742 + 0.3870967742 = 0.0000000000 0.0000000000 + x = 2.005196164 + 0.3870967742 x = 2.005196164 + 0.3870967742 Combine like terms: 2.005196164 + 0.3870967742 = 2.3922929382 x = 2.3922929382 Simplifying x = 2.3922929382Subproblem 2
x + -0.3870967742 = -2.005196164 Simplifying x + -0.3870967742 = -2.005196164 Reorder the terms: -0.3870967742 + x = -2.005196164 Solving -0.3870967742 + x = -2.005196164 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '0.3870967742' to each side of the equation. -0.3870967742 + 0.3870967742 + x = -2.005196164 + 0.3870967742 Combine like terms: -0.3870967742 + 0.3870967742 = 0.0000000000 0.0000000000 + x = -2.005196164 + 0.3870967742 x = -2.005196164 + 0.3870967742 Combine like terms: -2.005196164 + 0.3870967742 = -1.6180993898 x = -1.6180993898 Simplifying x = -1.6180993898Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.3922929382, -1.6180993898}
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