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Simplifying 27 = -16x2 + 96x + 3 Reorder the terms: 27 = 3 + 96x + -16x2 Solving 27 = 3 + 96x + -16x2 Solving for variable 'x'. Combine like terms: 27 + -3 = 24 24 + -96x + 16x2 = 3 + 96x + -16x2 + -3 + -96x + 16x2 Reorder the terms: 24 + -96x + 16x2 = 3 + -3 + 96x + -96x + -16x2 + 16x2 Combine like terms: 3 + -3 = 0 24 + -96x + 16x2 = 0 + 96x + -96x + -16x2 + 16x2 24 + -96x + 16x2 = 96x + -96x + -16x2 + 16x2 Combine like terms: 96x + -96x = 0 24 + -96x + 16x2 = 0 + -16x2 + 16x2 24 + -96x + 16x2 = -16x2 + 16x2 Combine like terms: -16x2 + 16x2 = 0 24 + -96x + 16x2 = 0 Factor out the Greatest Common Factor (GCF), '8'. 8(3 + -12x + 2x2) = 0 Ignore the factor 8.Subproblem 1
Set the factor '(3 + -12x + 2x2)' equal to zero and attempt to solve: Simplifying 3 + -12x + 2x2 = 0 Solving 3 + -12x + 2x2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 1.5 + -6x + x2 = 0 Move the constant term to the right: Add '-1.5' to each side of the equation. 1.5 + -6x + -1.5 + x2 = 0 + -1.5 Reorder the terms: 1.5 + -1.5 + -6x + x2 = 0 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + -6x + x2 = 0 + -1.5 -6x + x2 = 0 + -1.5 Combine like terms: 0 + -1.5 = -1.5 -6x + x2 = -1.5 The x term is -6x. Take half its coefficient (-3). Square it (9) and add it to both sides. Add '9' to each side of the equation. -6x + 9 + x2 = -1.5 + 9 Reorder the terms: 9 + -6x + x2 = -1.5 + 9 Combine like terms: -1.5 + 9 = 7.5 9 + -6x + x2 = 7.5 Factor a perfect square on the left side: (x + -3)(x + -3) = 7.5 Calculate the square root of the right side: 2.738612788 Break this problem into two subproblems by setting (x + -3) equal to 2.738612788 and -2.738612788.Subproblem 1
x + -3 = 2.738612788 Simplifying x + -3 = 2.738612788 Reorder the terms: -3 + x = 2.738612788 Solving -3 + x = 2.738612788 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + x = 2.738612788 + 3 Combine like terms: -3 + 3 = 0 0 + x = 2.738612788 + 3 x = 2.738612788 + 3 Combine like terms: 2.738612788 + 3 = 5.738612788 x = 5.738612788 Simplifying x = 5.738612788Subproblem 2
x + -3 = -2.738612788 Simplifying x + -3 = -2.738612788 Reorder the terms: -3 + x = -2.738612788 Solving -3 + x = -2.738612788 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '3' to each side of the equation. -3 + 3 + x = -2.738612788 + 3 Combine like terms: -3 + 3 = 0 0 + x = -2.738612788 + 3 x = -2.738612788 + 3 Combine like terms: -2.738612788 + 3 = 0.261387212 x = 0.261387212 Simplifying x = 0.261387212Solution
The solution to the problem is based on the solutions from the subproblems. x = {5.738612788, 0.261387212}Solution
x = {5.738612788, 0.261387212}
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