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Simplifying 180b2 + -180b + -60 = 0 Reorder the terms: -60 + -180b + 180b2 = 0 Solving -60 + -180b + 180b2 = 0 Solving for variable 'b'. Factor out the Greatest Common Factor (GCF), '60'. 60(-1 + -3b + 3b2) = 0 Ignore the factor 60.Subproblem 1
Set the factor '(-1 + -3b + 3b2)' equal to zero and attempt to solve: Simplifying -1 + -3b + 3b2 = 0 Solving -1 + -3b + 3b2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -0.3333333333 + -1b + b2 = 0 Move the constant term to the right: Add '0.3333333333' to each side of the equation. -0.3333333333 + -1b + 0.3333333333 + b2 = 0 + 0.3333333333 Reorder the terms: -0.3333333333 + 0.3333333333 + -1b + b2 = 0 + 0.3333333333 Combine like terms: -0.3333333333 + 0.3333333333 = 0.0000000000 0.0000000000 + -1b + b2 = 0 + 0.3333333333 -1b + b2 = 0 + 0.3333333333 Combine like terms: 0 + 0.3333333333 = 0.3333333333 -1b + b2 = 0.3333333333 The b term is -1b. 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. -1b + 0.25 + b2 = 0.3333333333 + 0.25 Reorder the terms: 0.25 + -1b + b2 = 0.3333333333 + 0.25 Combine like terms: 0.3333333333 + 0.25 = 0.5833333333 0.25 + -1b + b2 = 0.5833333333 Factor a perfect square on the left side: (b + -0.5)(b + -0.5) = 0.5833333333 Calculate the square root of the right side: 0.763762616 Break this problem into two subproblems by setting (b + -0.5) equal to 0.763762616 and -0.763762616.Subproblem 1
b + -0.5 = 0.763762616 Simplifying b + -0.5 = 0.763762616 Reorder the terms: -0.5 + b = 0.763762616 Solving -0.5 + b = 0.763762616 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + b = 0.763762616 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + b = 0.763762616 + 0.5 b = 0.763762616 + 0.5 Combine like terms: 0.763762616 + 0.5 = 1.263762616 b = 1.263762616 Simplifying b = 1.263762616Subproblem 2
b + -0.5 = -0.763762616 Simplifying b + -0.5 = -0.763762616 Reorder the terms: -0.5 + b = -0.763762616 Solving -0.5 + b = -0.763762616 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + b = -0.763762616 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + b = -0.763762616 + 0.5 b = -0.763762616 + 0.5 Combine like terms: -0.763762616 + 0.5 = -0.263762616 b = -0.263762616 Simplifying b = -0.263762616Solution
The solution to the problem is based on the solutions from the subproblems. b = {1.263762616, -0.263762616}Solution
b = {1.263762616, -0.263762616}
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