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Simplifying -8b2 + 4b = -8 Reorder the terms: 4b + -8b2 = -8 Solving 4b + -8b2 = -8 Solving for variable 'b'. Reorder the terms: 8 + 4b + -8b2 = -8 + 8 Combine like terms: -8 + 8 = 0 8 + 4b + -8b2 = 0 Factor out the Greatest Common Factor (GCF), '4'. 4(2 + b + -2b2) = 0 Ignore the factor 4.Subproblem 1
Set the factor '(2 + b + -2b2)' equal to zero and attempt to solve: Simplifying 2 + b + -2b2 = 0 Solving 2 + b + -2b2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -1 + -0.5b + b2 = 0 Move the constant term to the right: Add '1' to each side of the equation. -1 + -0.5b + 1 + b2 = 0 + 1 Reorder the terms: -1 + 1 + -0.5b + b2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + -0.5b + b2 = 0 + 1 -0.5b + b2 = 0 + 1 Combine like terms: 0 + 1 = 1 -0.5b + b2 = 1 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. -0.5b + 0.25 + b2 = 1 + 0.25 Reorder the terms: 0.25 + -0.5b + b2 = 1 + 0.25 Combine like terms: 1 + 0.25 = 1.25 0.25 + -0.5b + b2 = 1.25 Factor a perfect square on the left side: (b + 0.5)(b + 0.5) = 1.25 Calculate the square root of the right side: 1.118033989 Break this problem into two subproblems by setting (b + 0.5) equal to 1.118033989 and -1.118033989.Subproblem 1
b + 0.5 = 1.118033989 Simplifying b + 0.5 = 1.118033989 Reorder the terms: 0.5 + b = 1.118033989 Solving 0.5 + b = 1.118033989 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 = 1.118033989 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + b = 1.118033989 + -0.5 b = 1.118033989 + -0.5 Combine like terms: 1.118033989 + -0.5 = 0.618033989 b = 0.618033989 Simplifying b = 0.618033989Subproblem 2
b + 0.5 = -1.118033989 Simplifying b + 0.5 = -1.118033989 Reorder the terms: 0.5 + b = -1.118033989 Solving 0.5 + b = -1.118033989 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 = -1.118033989 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + b = -1.118033989 + -0.5 b = -1.118033989 + -0.5 Combine like terms: -1.118033989 + -0.5 = -1.618033989 b = -1.618033989 Simplifying b = -1.618033989Solution
The solution to the problem is based on the solutions from the subproblems. b = {0.618033989, -1.618033989}Solution
b = {0.618033989, -1.618033989}
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