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Simplifying 240x3 + 180x2 + -1440x = 0 Reorder the terms: -1440x + 180x2 + 240x3 = 0 Solving -1440x + 180x2 + 240x3 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), '60x'. 60x(-24 + 3x + 4x2) = 0 Ignore the factor 60.Subproblem 1
Set the factor 'x' equal to zero and attempt to solve: Simplifying x = 0 Solving x = 0 Move all terms containing x to the left, all other terms to the right. Simplifying x = 0Subproblem 2
Set the factor '(-24 + 3x + 4x2)' equal to zero and attempt to solve: Simplifying -24 + 3x + 4x2 = 0 Solving -24 + 3x + 4x2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -6 + 0.75x + x2 = 0 Move the constant term to the right: Add '6' to each side of the equation. -6 + 0.75x + 6 + x2 = 0 + 6 Reorder the terms: -6 + 6 + 0.75x + x2 = 0 + 6 Combine like terms: -6 + 6 = 0 0 + 0.75x + x2 = 0 + 6 0.75x + x2 = 0 + 6 Combine like terms: 0 + 6 = 6 0.75x + x2 = 6 The x term is 0.75x. Take half its coefficient (0.375). Square it (0.140625) and add it to both sides. Add '0.140625' to each side of the equation. 0.75x + 0.140625 + x2 = 6 + 0.140625 Reorder the terms: 0.140625 + 0.75x + x2 = 6 + 0.140625 Combine like terms: 6 + 0.140625 = 6.140625 0.140625 + 0.75x + x2 = 6.140625 Factor a perfect square on the left side: (x + 0.375)(x + 0.375) = 6.140625 Calculate the square root of the right side: 2.47802845 Break this problem into two subproblems by setting (x + 0.375) equal to 2.47802845 and -2.47802845.Subproblem 1
x + 0.375 = 2.47802845 Simplifying x + 0.375 = 2.47802845 Reorder the terms: 0.375 + x = 2.47802845 Solving 0.375 + x = 2.47802845 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.375' to each side of the equation. 0.375 + -0.375 + x = 2.47802845 + -0.375 Combine like terms: 0.375 + -0.375 = 0.000 0.000 + x = 2.47802845 + -0.375 x = 2.47802845 + -0.375 Combine like terms: 2.47802845 + -0.375 = 2.10302845 x = 2.10302845 Simplifying x = 2.10302845Subproblem 2
x + 0.375 = -2.47802845 Simplifying x + 0.375 = -2.47802845 Reorder the terms: 0.375 + x = -2.47802845 Solving 0.375 + x = -2.47802845 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.375' to each side of the equation. 0.375 + -0.375 + x = -2.47802845 + -0.375 Combine like terms: 0.375 + -0.375 = 0.000 0.000 + x = -2.47802845 + -0.375 x = -2.47802845 + -0.375 Combine like terms: -2.47802845 + -0.375 = -2.85302845 x = -2.85302845 Simplifying x = -2.85302845Solution
The solution to the problem is based on the solutions from the subproblems. x = {2.10302845, -2.85302845}Solution
x = {0, 2.10302845, -2.85302845}
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