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Simplifying 4b2 + 8b + 4b2 = 4 Reorder the terms: 8b + 4b2 + 4b2 = 4 Combine like terms: 4b2 + 4b2 = 8b2 8b + 8b2 = 4 Solving 8b + 8b2 = 4 Solving for variable 'b'. Reorder the terms: -4 + 8b + 8b2 = 4 + -4 Combine like terms: 4 + -4 = 0 -4 + 8b + 8b2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(-1 + 2b + 2b2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(-1 + 2b + 2b2)' equal to zero and attempt to solve: Simplifying -1 + 2b + 2b2 = 0 Solving -1 + 2b + 2b2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -0.5 + b + b2 = 0 Move the constant term to the right: Add '0.5' to each side of the equation. -0.5 + b + 0.5 + b2 = 0 + 0.5 Reorder the terms: -0.5 + 0.5 + b + b2 = 0 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + b + b2 = 0 + 0.5 b + b2 = 0 + 0.5 Combine like terms: 0 + 0.5 = 0.5 b + b2 = 0.5 The b term is b. 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. + 0.25 + b2 = 0.5 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + b2 = 0.5 + 0.25 Combine like terms: 0.5 + 0.25 = 0.75 1.25 + b2 = 0.75 Factor a perfect square on the left side: (b + 0.5)(b + 0.5) = 0.75 Calculate the square root of the right side: 0.866025404 Break this problem into two subproblems by setting (b + 0.5) equal to 0.866025404 and -0.866025404.Subproblem 1
b + 0.5 = 0.866025404 Simplifying b + 0.5 = 0.866025404 Reorder the terms: 0.5 + b = 0.866025404 Solving 0.5 + b = 0.866025404 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 = 0.866025404 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + b = 0.866025404 + -0.5 b = 0.866025404 + -0.5 Combine like terms: 0.866025404 + -0.5 = 0.366025404 b = 0.366025404 Simplifying b = 0.366025404Subproblem 2
b + 0.5 = -0.866025404 Simplifying b + 0.5 = -0.866025404 Reorder the terms: 0.5 + b = -0.866025404 Solving 0.5 + b = -0.866025404 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 = -0.866025404 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + b = -0.866025404 + -0.5 b = -0.866025404 + -0.5 Combine like terms: -0.866025404 + -0.5 = -1.366025404 b = -1.366025404 Simplifying b = -1.366025404Solution
The solution to the problem is based on the solutions from the subproblems. b = {0.366025404, -1.366025404}Solution
b = {0.366025404, -1.366025404}
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