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Simplifying 120x2 + 120x + -40 = 0 Reorder the terms: -40 + 120x + 120x2 = 0 Solving -40 + 120x + 120x2 = 0 Solving for variable 'x'. Factor out the Greatest Common Factor (GCF), '40'. 40(-1 + 3x + 3x2) = 0 Ignore the factor 40.Subproblem 1
Set the factor '(-1 + 3x + 3x2)' equal to zero and attempt to solve: Simplifying -1 + 3x + 3x2 = 0 Solving -1 + 3x + 3x2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -0.3333333333 + x + x2 = 0 Move the constant term to the right: Add '0.3333333333' to each side of the equation. -0.3333333333 + x + 0.3333333333 + x2 = 0 + 0.3333333333 Reorder the terms: -0.3333333333 + 0.3333333333 + x + x2 = 0 + 0.3333333333 Combine like terms: -0.3333333333 + 0.3333333333 = 0.0000000000 0.0000000000 + x + x2 = 0 + 0.3333333333 x + x2 = 0 + 0.3333333333 Combine like terms: 0 + 0.3333333333 = 0.3333333333 x + x2 = 0.3333333333 The x term is x. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. + 0.25 + x2 = 0.3333333333 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + x2 = 0.3333333333 + 0.25 Combine like terms: 0.3333333333 + 0.25 = 0.5833333333 1.25 + x2 = 0.5833333333 Factor a perfect square on the left side: (x + 0.5)(x + 0.5) = 0.5833333333 Calculate the square root of the right side: 0.763762616 Break this problem into two subproblems by setting (x + 0.5) equal to 0.763762616 and -0.763762616.Subproblem 1
x + 0.5 = 0.763762616 Simplifying x + 0.5 = 0.763762616 Reorder the terms: 0.5 + x = 0.763762616 Solving 0.5 + x = 0.763762616 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = 0.763762616 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = 0.763762616 + -0.5 x = 0.763762616 + -0.5 Combine like terms: 0.763762616 + -0.5 = 0.263762616 x = 0.263762616 Simplifying x = 0.263762616Subproblem 2
x + 0.5 = -0.763762616 Simplifying x + 0.5 = -0.763762616 Reorder the terms: 0.5 + x = -0.763762616 Solving 0.5 + x = -0.763762616 Solving for variable 'x'. Move all terms containing x to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + x = -0.763762616 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + x = -0.763762616 + -0.5 x = -0.763762616 + -0.5 Combine like terms: -0.763762616 + -0.5 = -1.263762616 x = -1.263762616 Simplifying x = -1.263762616Solution
The solution to the problem is based on the solutions from the subproblems. x = {0.263762616, -1.263762616}Solution
x = {0.263762616, -1.263762616}
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