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Simplifying 2b3 + -4b2 = 2b Reorder the terms: -4b2 + 2b3 = 2b Solving -4b2 + 2b3 = 2b Solving for variable 'b'. Reorder the terms: -2b + -4b2 + 2b3 = 2b + -2b Combine like terms: 2b + -2b = 0 -2b + -4b2 + 2b3 = 0 Factor out the Greatest Common Factor (GCF), '2b'. 2b(-1 + -2b + 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 + -2b + b2)' equal to zero and attempt to solve: Simplifying -1 + -2b + b2 = 0 Solving -1 + -2b + b2 = 0 Begin completing the square. Move the constant term to the right: Add '1' to each side of the equation. -1 + -2b + 1 + b2 = 0 + 1 Reorder the terms: -1 + 1 + -2b + b2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + -2b + b2 = 0 + 1 -2b + b2 = 0 + 1 Combine like terms: 0 + 1 = 1 -2b + b2 = 1 The b term is -2b. Take half its coefficient (-1). Square it (1) and add it to both sides. Add '1' to each side of the equation. -2b + 1 + b2 = 1 + 1 Reorder the terms: 1 + -2b + b2 = 1 + 1 Combine like terms: 1 + 1 = 2 1 + -2b + b2 = 2 Factor a perfect square on the left side: (b + -1)(b + -1) = 2 Calculate the square root of the right side: 1.414213562 Break this problem into two subproblems by setting (b + -1) equal to 1.414213562 and -1.414213562.Subproblem 1
b + -1 = 1.414213562 Simplifying b + -1 = 1.414213562 Reorder the terms: -1 + b = 1.414213562 Solving -1 + b = 1.414213562 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + b = 1.414213562 + 1 Combine like terms: -1 + 1 = 0 0 + b = 1.414213562 + 1 b = 1.414213562 + 1 Combine like terms: 1.414213562 + 1 = 2.414213562 b = 2.414213562 Simplifying b = 2.414213562Subproblem 2
b + -1 = -1.414213562 Simplifying b + -1 = -1.414213562 Reorder the terms: -1 + b = -1.414213562 Solving -1 + b = -1.414213562 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + b = -1.414213562 + 1 Combine like terms: -1 + 1 = 0 0 + b = -1.414213562 + 1 b = -1.414213562 + 1 Combine like terms: -1.414213562 + 1 = -0.414213562 b = -0.414213562 Simplifying b = -0.414213562Solution
The solution to the problem is based on the solutions from the subproblems. b = {2.414213562, -0.414213562}Solution
b = {0, 2.414213562, -0.414213562}
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