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Simplifying 9b2 = -12 + -27b Solving 9b2 = -12 + -27b Solving for variable 'b'. Reorder the terms: 12 + 27b + 9b2 = -12 + -27b + 12 + 27b Reorder the terms: 12 + 27b + 9b2 = -12 + 12 + -27b + 27b Combine like terms: -12 + 12 = 0 12 + 27b + 9b2 = 0 + -27b + 27b 12 + 27b + 9b2 = -27b + 27b Combine like terms: -27b + 27b = 0 12 + 27b + 9b2 = 0 Factor out the Greatest Common Factor (GCF), '3'. 3(4 + 9b + 3b2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(4 + 9b + 3b2)' equal to zero and attempt to solve: Simplifying 4 + 9b + 3b2 = 0 Solving 4 + 9b + 3b2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. 1.333333333 + 3b + b2 = 0 Move the constant term to the right: Add '-1.333333333' to each side of the equation. 1.333333333 + 3b + -1.333333333 + b2 = 0 + -1.333333333 Reorder the terms: 1.333333333 + -1.333333333 + 3b + b2 = 0 + -1.333333333 Combine like terms: 1.333333333 + -1.333333333 = 0.000000000 0.000000000 + 3b + b2 = 0 + -1.333333333 3b + b2 = 0 + -1.333333333 Combine like terms: 0 + -1.333333333 = -1.333333333 3b + b2 = -1.333333333 The b term is 3b. 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. 3b + 2.25 + b2 = -1.333333333 + 2.25 Reorder the terms: 2.25 + 3b + b2 = -1.333333333 + 2.25 Combine like terms: -1.333333333 + 2.25 = 0.916666667 2.25 + 3b + b2 = 0.916666667 Factor a perfect square on the left side: (b + 1.5)(b + 1.5) = 0.916666667 Calculate the square root of the right side: 0.957427108 Break this problem into two subproblems by setting (b + 1.5) equal to 0.957427108 and -0.957427108.Subproblem 1
b + 1.5 = 0.957427108 Simplifying b + 1.5 = 0.957427108 Reorder the terms: 1.5 + b = 0.957427108 Solving 1.5 + b = 0.957427108 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + b = 0.957427108 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + b = 0.957427108 + -1.5 b = 0.957427108 + -1.5 Combine like terms: 0.957427108 + -1.5 = -0.542572892 b = -0.542572892 Simplifying b = -0.542572892Subproblem 2
b + 1.5 = -0.957427108 Simplifying b + 1.5 = -0.957427108 Reorder the terms: 1.5 + b = -0.957427108 Solving 1.5 + b = -0.957427108 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + b = -0.957427108 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + b = -0.957427108 + -1.5 b = -0.957427108 + -1.5 Combine like terms: -0.957427108 + -1.5 = -2.457427108 b = -2.457427108 Simplifying b = -2.457427108Solution
The solution to the problem is based on the solutions from the subproblems. b = {-0.542572892, -2.457427108}Solution
b = {-0.542572892, -2.457427108}
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