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Simplifying -5b + 24 = -8b(b + -6) + 6b Reorder the terms: 24 + -5b = -8b(b + -6) + 6b Reorder the terms: 24 + -5b = -8b(-6 + b) + 6b 24 + -5b = (-6 * -8b + b * -8b) + 6b 24 + -5b = (48b + -8b2) + 6b Reorder the terms: 24 + -5b = 48b + 6b + -8b2 Combine like terms: 48b + 6b = 54b 24 + -5b = 54b + -8b2 Solving 24 + -5b = 54b + -8b2 Solving for variable 'b'. Combine like terms: -5b + -54b = -59b 24 + -59b + 8b2 = 54b + -8b2 + -54b + 8b2 Reorder the terms: 24 + -59b + 8b2 = 54b + -54b + -8b2 + 8b2 Combine like terms: 54b + -54b = 0 24 + -59b + 8b2 = 0 + -8b2 + 8b2 24 + -59b + 8b2 = -8b2 + 8b2 Combine like terms: -8b2 + 8b2 = 0 24 + -59b + 8b2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. 3 + -7.375b + b2 = 0 Move the constant term to the right: Add '-3' to each side of the equation. 3 + -7.375b + -3 + b2 = 0 + -3 Reorder the terms: 3 + -3 + -7.375b + b2 = 0 + -3 Combine like terms: 3 + -3 = 0 0 + -7.375b + b2 = 0 + -3 -7.375b + b2 = 0 + -3 Combine like terms: 0 + -3 = -3 -7.375b + b2 = -3 The b term is -7.375b. Take half its coefficient (-3.6875). Square it (13.59765625) and add it to both sides. Add '13.59765625' to each side of the equation. -7.375b + 13.59765625 + b2 = -3 + 13.59765625 Reorder the terms: 13.59765625 + -7.375b + b2 = -3 + 13.59765625 Combine like terms: -3 + 13.59765625 = 10.59765625 13.59765625 + -7.375b + b2 = 10.59765625 Factor a perfect square on the left side: (b + -3.6875)(b + -3.6875) = 10.59765625 Calculate the square root of the right side: 3.255404161 Break this problem into two subproblems by setting (b + -3.6875) equal to 3.255404161 and -3.255404161.Subproblem 1
b + -3.6875 = 3.255404161 Simplifying b + -3.6875 = 3.255404161 Reorder the terms: -3.6875 + b = 3.255404161 Solving -3.6875 + b = 3.255404161 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '3.6875' to each side of the equation. -3.6875 + 3.6875 + b = 3.255404161 + 3.6875 Combine like terms: -3.6875 + 3.6875 = 0.0000 0.0000 + b = 3.255404161 + 3.6875 b = 3.255404161 + 3.6875 Combine like terms: 3.255404161 + 3.6875 = 6.942904161 b = 6.942904161 Simplifying b = 6.942904161Subproblem 2
b + -3.6875 = -3.255404161 Simplifying b + -3.6875 = -3.255404161 Reorder the terms: -3.6875 + b = -3.255404161 Solving -3.6875 + b = -3.255404161 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '3.6875' to each side of the equation. -3.6875 + 3.6875 + b = -3.255404161 + 3.6875 Combine like terms: -3.6875 + 3.6875 = 0.0000 0.0000 + b = -3.255404161 + 3.6875 b = -3.255404161 + 3.6875 Combine like terms: -3.255404161 + 3.6875 = 0.432095839 b = 0.432095839 Simplifying b = 0.432095839Solution
The solution to the problem is based on the solutions from the subproblems. b = {6.942904161, 0.432095839}
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