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Simplifying 8b2 + 6b + -192 = 0 Reorder the terms: -192 + 6b + 8b2 = 0 Solving -192 + 6b + 8b2 = 0 Solving for variable 'b'. Factor out the Greatest Common Factor (GCF), '2'. 2(-96 + 3b + 4b2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-96 + 3b + 4b2)' equal to zero and attempt to solve: Simplifying -96 + 3b + 4b2 = 0 Solving -96 + 3b + 4b2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -24 + 0.75b + b2 = 0 Move the constant term to the right: Add '24' to each side of the equation. -24 + 0.75b + 24 + b2 = 0 + 24 Reorder the terms: -24 + 24 + 0.75b + b2 = 0 + 24 Combine like terms: -24 + 24 = 0 0 + 0.75b + b2 = 0 + 24 0.75b + b2 = 0 + 24 Combine like terms: 0 + 24 = 24 0.75b + b2 = 24 The b term is 0.75b. Take half its coefficient (0.375). Square it (0.140625) and add it to both sides. Add '0.140625' to each side of the equation. 0.75b + 0.140625 + b2 = 24 + 0.140625 Reorder the terms: 0.140625 + 0.75b + b2 = 24 + 0.140625 Combine like terms: 24 + 0.140625 = 24.140625 0.140625 + 0.75b + b2 = 24.140625 Factor a perfect square on the left side: (b + 0.375)(b + 0.375) = 24.140625 Calculate the square root of the right side: 4.913311002 Break this problem into two subproblems by setting (b + 0.375) equal to 4.913311002 and -4.913311002.Subproblem 1
b + 0.375 = 4.913311002 Simplifying b + 0.375 = 4.913311002 Reorder the terms: 0.375 + b = 4.913311002 Solving 0.375 + b = 4.913311002 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-0.375' to each side of the equation. 0.375 + -0.375 + b = 4.913311002 + -0.375 Combine like terms: 0.375 + -0.375 = 0.000 0.000 + b = 4.913311002 + -0.375 b = 4.913311002 + -0.375 Combine like terms: 4.913311002 + -0.375 = 4.538311002 b = 4.538311002 Simplifying b = 4.538311002Subproblem 2
b + 0.375 = -4.913311002 Simplifying b + 0.375 = -4.913311002 Reorder the terms: 0.375 + b = -4.913311002 Solving 0.375 + b = -4.913311002 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-0.375' to each side of the equation. 0.375 + -0.375 + b = -4.913311002 + -0.375 Combine like terms: 0.375 + -0.375 = 0.000 0.000 + b = -4.913311002 + -0.375 b = -4.913311002 + -0.375 Combine like terms: -4.913311002 + -0.375 = -5.288311002 b = -5.288311002 Simplifying b = -5.288311002Solution
The solution to the problem is based on the solutions from the subproblems. b = {4.538311002, -5.288311002}Solution
b = {4.538311002, -5.288311002}
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