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Simplifying 0 = 2x2 + 6x + -54 Reorder the terms: 0 = -54 + 6x + 2x2 Solving 0 = -54 + 6x + 2x2 Solving for variable 'x'. Combine like terms: 0 + 54 = 54 54 + -6x + -2x2 = -54 + 6x + 2x2 + 54 + -6x + -2x2 Reorder the terms: 54 + -6x + -2x2 = -54 + 54 + 6x + -6x + 2x2 + -2x2 Combine like terms: -54 + 54 = 0 54 + -6x + -2x2 = 0 + 6x + -6x + 2x2 + -2x2 54 + -6x + -2x2 = 6x + -6x + 2x2 + -2x2 Combine like terms: 6x + -6x = 0 54 + -6x + -2x2 = 0 + 2x2 + -2x2 54 + -6x + -2x2 = 2x2 + -2x2 Combine like terms: 2x2 + -2x2 = 0 54 + -6x + -2x2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(27 + -3x + -1x2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(27 + -3x + -1x2)' equal to zero and attempt to solve: Simplifying 27 + -3x + -1x2 = 0 Solving 27 + -3x + -1x2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -27 + 3x + x2 = 0 Move the constant term to the right: Add '27' to each side of the equation. -27 + 3x + 27 + x2 = 0 + 27 Reorder the terms: -27 + 27 + 3x + x2 = 0 + 27 Combine like terms: -27 + 27 = 0 0 + 3x + x2 = 0 + 27 3x + x2 = 0 + 27 Combine like terms: 0 + 27 = 27 3x + x2 = 27 The x term is 3x. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3x + 2.25 + x2 = 27 + 2.25 Reorder the terms: 2.25 + 3x + x2 = 27 + 2.25 Combine like terms: 27 + 2.25 = 29.25 2.25 + 3x + x2 = 29.25 Factor a perfect square on the left side: (x + 1.5)(x + 1.5) = 29.25 Calculate the square root of the right side: 5.408326913 Break this problem into two subproblems by setting (x + 1.5) equal to 5.408326913 and -5.408326913.Subproblem 1
x + 1.5 = 5.408326913 Simplifying x + 1.5 = 5.408326913 Reorder the terms: 1.5 + x = 5.408326913 Solving 1.5 + x = 5.408326913 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + x = 5.408326913 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = 5.408326913 + -1.5 x = 5.408326913 + -1.5 Combine like terms: 5.408326913 + -1.5 = 3.908326913 x = 3.908326913 Simplifying x = 3.908326913Subproblem 2
x + 1.5 = -5.408326913 Simplifying x + 1.5 = -5.408326913 Reorder the terms: 1.5 + x = -5.408326913 Solving 1.5 + x = -5.408326913 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + x = -5.408326913 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + x = -5.408326913 + -1.5 x = -5.408326913 + -1.5 Combine like terms: -5.408326913 + -1.5 = -6.908326913 x = -6.908326913 Simplifying x = -6.908326913Solution
The solution to the problem is based on the solutions from the subproblems. x = {3.908326913, -6.908326913}Solution
x = {3.908326913, -6.908326913}
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