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Simplifying 9b2 + -9b + -12 = 0 Reorder the terms: -12 + -9b + 9b2 = 0 Solving -12 + -9b + 9b2 = 0 Solving for variable 'b'. Factor out the Greatest Common Factor (GCF), '3'. 3(-4 + -3b + 3b2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(-4 + -3b + 3b2)' equal to zero and attempt to solve: Simplifying -4 + -3b + 3b2 = 0 Solving -4 + -3b + 3b2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -1.333333333 + -1b + b2 = 0 Move the constant term to the right: Add '1.333333333' to each side of the equation. -1.333333333 + -1b + 1.333333333 + b2 = 0 + 1.333333333 Reorder the terms: -1.333333333 + 1.333333333 + -1b + b2 = 0 + 1.333333333 Combine like terms: -1.333333333 + 1.333333333 = 0.000000000 0.000000000 + -1b + b2 = 0 + 1.333333333 -1b + b2 = 0 + 1.333333333 Combine like terms: 0 + 1.333333333 = 1.333333333 -1b + b2 = 1.333333333 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 = 1.333333333 + 0.25 Reorder the terms: 0.25 + -1b + b2 = 1.333333333 + 0.25 Combine like terms: 1.333333333 + 0.25 = 1.583333333 0.25 + -1b + b2 = 1.583333333 Factor a perfect square on the left side: (b + -0.5)(b + -0.5) = 1.583333333 Calculate the square root of the right side: 1.258305739 Break this problem into two subproblems by setting (b + -0.5) equal to 1.258305739 and -1.258305739.Subproblem 1
b + -0.5 = 1.258305739 Simplifying b + -0.5 = 1.258305739 Reorder the terms: -0.5 + b = 1.258305739 Solving -0.5 + b = 1.258305739 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 = 1.258305739 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + b = 1.258305739 + 0.5 b = 1.258305739 + 0.5 Combine like terms: 1.258305739 + 0.5 = 1.758305739 b = 1.758305739 Simplifying b = 1.758305739Subproblem 2
b + -0.5 = -1.258305739 Simplifying b + -0.5 = -1.258305739 Reorder the terms: -0.5 + b = -1.258305739 Solving -0.5 + b = -1.258305739 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 = -1.258305739 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + b = -1.258305739 + 0.5 b = -1.258305739 + 0.5 Combine like terms: -1.258305739 + 0.5 = -0.758305739 b = -0.758305739 Simplifying b = -0.758305739Solution
The solution to the problem is based on the solutions from the subproblems. b = {1.758305739, -0.758305739}Solution
b = {1.758305739, -0.758305739}
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