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Simplifying 36b3 + -30b2 + 4b = 2b Reorder the terms: 4b + -30b2 + 36b3 = 2b Solving 4b + -30b2 + 36b3 = 2b Solving for variable 'b'. Reorder the terms: 4b + -2b + -30b2 + 36b3 = 2b + -2b Combine like terms: 4b + -2b = 2b 2b + -30b2 + 36b3 = 2b + -2b Combine like terms: 2b + -2b = 0 2b + -30b2 + 36b3 = 0 Factor out the Greatest Common Factor (GCF), '2b'. 2b(1 + -15b + 18b2) = 0 Ignore the factor 2.Subproblem 1
Set the factor 'b' equal to zero and attempt to solve: Simplifying b = 0 Solving b = 0 Move all terms containing b to the left, all other terms to the right. Simplifying b = 0Subproblem 2
Set the factor '(1 + -15b + 18b2)' equal to zero and attempt to solve: Simplifying 1 + -15b + 18b2 = 0 Solving 1 + -15b + 18b2 = 0 Begin completing the square. Divide all terms by 18 the coefficient of the squared term: Divide each side by '18'. 0.05555555556 + -0.8333333333b + b2 = 0 Move the constant term to the right: Add '-0.05555555556' to each side of the equation. 0.05555555556 + -0.8333333333b + -0.05555555556 + b2 = 0 + -0.05555555556 Reorder the terms: 0.05555555556 + -0.05555555556 + -0.8333333333b + b2 = 0 + -0.05555555556 Combine like terms: 0.05555555556 + -0.05555555556 = 0.00000000000 0.00000000000 + -0.8333333333b + b2 = 0 + -0.05555555556 -0.8333333333b + b2 = 0 + -0.05555555556 Combine like terms: 0 + -0.05555555556 = -0.05555555556 -0.8333333333b + b2 = -0.05555555556 The b term is -0.8333333333b. Take half its coefficient (-0.4166666667). Square it (0.1736111111) and add it to both sides. Add '0.1736111111' to each side of the equation. -0.8333333333b + 0.1736111111 + b2 = -0.05555555556 + 0.1736111111 Reorder the terms: 0.1736111111 + -0.8333333333b + b2 = -0.05555555556 + 0.1736111111 Combine like terms: -0.05555555556 + 0.1736111111 = 0.11805555554 0.1736111111 + -0.8333333333b + b2 = 0.11805555554 Factor a perfect square on the left side: (b + -0.4166666667)(b + -0.4166666667) = 0.11805555554 Calculate the square root of the right side: 0.343592135 Break this problem into two subproblems by setting (b + -0.4166666667) equal to 0.343592135 and -0.343592135.Subproblem 1
b + -0.4166666667 = 0.343592135 Simplifying b + -0.4166666667 = 0.343592135 Reorder the terms: -0.4166666667 + b = 0.343592135 Solving -0.4166666667 + b = 0.343592135 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '0.4166666667' to each side of the equation. -0.4166666667 + 0.4166666667 + b = 0.343592135 + 0.4166666667 Combine like terms: -0.4166666667 + 0.4166666667 = 0.0000000000 0.0000000000 + b = 0.343592135 + 0.4166666667 b = 0.343592135 + 0.4166666667 Combine like terms: 0.343592135 + 0.4166666667 = 0.7602588017 b = 0.7602588017 Simplifying b = 0.7602588017Subproblem 2
b + -0.4166666667 = -0.343592135 Simplifying b + -0.4166666667 = -0.343592135 Reorder the terms: -0.4166666667 + b = -0.343592135 Solving -0.4166666667 + b = -0.343592135 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '0.4166666667' to each side of the equation. -0.4166666667 + 0.4166666667 + b = -0.343592135 + 0.4166666667 Combine like terms: -0.4166666667 + 0.4166666667 = 0.0000000000 0.0000000000 + b = -0.343592135 + 0.4166666667 b = -0.343592135 + 0.4166666667 Combine like terms: -0.343592135 + 0.4166666667 = 0.0730745317 b = 0.0730745317 Simplifying b = 0.0730745317Solution
The solution to the problem is based on the solutions from the subproblems. b = {0.7602588017, 0.0730745317}Solution
b = {0, 0.7602588017, 0.0730745317}
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