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Simplifying 9b2 + -48b + 48 + 15b2 + -20b = 20b + -5b2 Reorder the terms: 48 + -48b + -20b + 9b2 + 15b2 = 20b + -5b2 Combine like terms: -48b + -20b = -68b 48 + -68b + 9b2 + 15b2 = 20b + -5b2 Combine like terms: 9b2 + 15b2 = 24b2 48 + -68b + 24b2 = 20b + -5b2 Solving 48 + -68b + 24b2 = 20b + -5b2 Solving for variable 'b'. Reorder the terms: 48 + -68b + -20b + 24b2 + 5b2 = 20b + -5b2 + -20b + 5b2 Combine like terms: -68b + -20b = -88b 48 + -88b + 24b2 + 5b2 = 20b + -5b2 + -20b + 5b2 Combine like terms: 24b2 + 5b2 = 29b2 48 + -88b + 29b2 = 20b + -5b2 + -20b + 5b2 Reorder the terms: 48 + -88b + 29b2 = 20b + -20b + -5b2 + 5b2 Combine like terms: 20b + -20b = 0 48 + -88b + 29b2 = 0 + -5b2 + 5b2 48 + -88b + 29b2 = -5b2 + 5b2 Combine like terms: -5b2 + 5b2 = 0 48 + -88b + 29b2 = 0 Begin completing the square. Divide all terms by 29 the coefficient of the squared term: Divide each side by '29'. 1.655172414 + -3.034482759b + b2 = 0 Move the constant term to the right: Add '-1.655172414' to each side of the equation. 1.655172414 + -3.034482759b + -1.655172414 + b2 = 0 + -1.655172414 Reorder the terms: 1.655172414 + -1.655172414 + -3.034482759b + b2 = 0 + -1.655172414 Combine like terms: 1.655172414 + -1.655172414 = 0.000000000 0.000000000 + -3.034482759b + b2 = 0 + -1.655172414 -3.034482759b + b2 = 0 + -1.655172414 Combine like terms: 0 + -1.655172414 = -1.655172414 -3.034482759b + b2 = -1.655172414 The b term is -3.034482759b. Take half its coefficient (-1.51724138). Square it (2.302021405) and add it to both sides. Add '2.302021405' to each side of the equation. -3.034482759b + 2.302021405 + b2 = -1.655172414 + 2.302021405 Reorder the terms: 2.302021405 + -3.034482759b + b2 = -1.655172414 + 2.302021405 Combine like terms: -1.655172414 + 2.302021405 = 0.646848991 2.302021405 + -3.034482759b + b2 = 0.646848991 Factor a perfect square on the left side: (b + -1.51724138)(b + -1.51724138) = 0.646848991 Calculate the square root of the right side: 0.804269228 Break this problem into two subproblems by setting (b + -1.51724138) equal to 0.804269228 and -0.804269228.Subproblem 1
b + -1.51724138 = 0.804269228 Simplifying b + -1.51724138 = 0.804269228 Reorder the terms: -1.51724138 + b = 0.804269228 Solving -1.51724138 + b = 0.804269228 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '1.51724138' to each side of the equation. -1.51724138 + 1.51724138 + b = 0.804269228 + 1.51724138 Combine like terms: -1.51724138 + 1.51724138 = 0.00000000 0.00000000 + b = 0.804269228 + 1.51724138 b = 0.804269228 + 1.51724138 Combine like terms: 0.804269228 + 1.51724138 = 2.321510608 b = 2.321510608 Simplifying b = 2.321510608Subproblem 2
b + -1.51724138 = -0.804269228 Simplifying b + -1.51724138 = -0.804269228 Reorder the terms: -1.51724138 + b = -0.804269228 Solving -1.51724138 + b = -0.804269228 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '1.51724138' to each side of the equation. -1.51724138 + 1.51724138 + b = -0.804269228 + 1.51724138 Combine like terms: -1.51724138 + 1.51724138 = 0.00000000 0.00000000 + b = -0.804269228 + 1.51724138 b = -0.804269228 + 1.51724138 Combine like terms: -0.804269228 + 1.51724138 = 0.712972152 b = 0.712972152 Simplifying b = 0.712972152Solution
The solution to the problem is based on the solutions from the subproblems. b = {2.321510608, 0.712972152}
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