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Simplifying 2b(b2 + 3b + 1) = 0 Reorder the terms: 2b(1 + 3b + b2) = 0 (1 * 2b + 3b * 2b + b2 * 2b) = 0 (2b + 6b2 + 2b3) = 0 Solving 2b + 6b2 + 2b3 = 0 Solving for variable 'b'. Factor out the Greatest Common Factor (GCF), '2b'. 2b(1 + 3b + b2) = 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 + 3b + b2)' equal to zero and attempt to solve: Simplifying 1 + 3b + b2 = 0 Solving 1 + 3b + b2 = 0 Begin completing the square. Move the constant term to the right: Add '-1' to each side of the equation. 1 + 3b + -1 + b2 = 0 + -1 Reorder the terms: 1 + -1 + 3b + b2 = 0 + -1 Combine like terms: 1 + -1 = 0 0 + 3b + b2 = 0 + -1 3b + b2 = 0 + -1 Combine like terms: 0 + -1 = -1 3b + b2 = -1 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 + 2.25 Reorder the terms: 2.25 + 3b + b2 = -1 + 2.25 Combine like terms: -1 + 2.25 = 1.25 2.25 + 3b + b2 = 1.25 Factor a perfect square on the left side: (b + 1.5)(b + 1.5) = 1.25 Calculate the square root of the right side: 1.118033989 Break this problem into two subproblems by setting (b + 1.5) equal to 1.118033989 and -1.118033989.Subproblem 1
b + 1.5 = 1.118033989 Simplifying b + 1.5 = 1.118033989 Reorder the terms: 1.5 + b = 1.118033989 Solving 1.5 + b = 1.118033989 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 = 1.118033989 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + b = 1.118033989 + -1.5 b = 1.118033989 + -1.5 Combine like terms: 1.118033989 + -1.5 = -0.381966011 b = -0.381966011 Simplifying b = -0.381966011Subproblem 2
b + 1.5 = -1.118033989 Simplifying b + 1.5 = -1.118033989 Reorder the terms: 1.5 + b = -1.118033989 Solving 1.5 + b = -1.118033989 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 = -1.118033989 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + b = -1.118033989 + -1.5 b = -1.118033989 + -1.5 Combine like terms: -1.118033989 + -1.5 = -2.618033989 b = -2.618033989 Simplifying b = -2.618033989Solution
The solution to the problem is based on the solutions from the subproblems. b = {-0.381966011, -2.618033989}Solution
b = {0, -0.381966011, -2.618033989}
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