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Simplifying (b + -1) + b(b + 2) = 11 Reorder the terms: (-1 + b) + b(b + 2) = 11 Remove parenthesis around (-1 + b) -1 + b + b(b + 2) = 11 Reorder the terms: -1 + b + b(2 + b) = 11 -1 + b + (2 * b + b * b) = 11 -1 + b + (2b + b2) = 11 Combine like terms: b + 2b = 3b -1 + 3b + b2 = 11 Solving -1 + 3b + b2 = 11 Solving for variable 'b'. Reorder the terms: -1 + -11 + 3b + b2 = 11 + -11 Combine like terms: -1 + -11 = -12 -12 + 3b + b2 = 11 + -11 Combine like terms: 11 + -11 = 0 -12 + 3b + b2 = 0 Begin completing the square. Move the constant term to the right: Add '12' to each side of the equation. -12 + 3b + 12 + b2 = 0 + 12 Reorder the terms: -12 + 12 + 3b + b2 = 0 + 12 Combine like terms: -12 + 12 = 0 0 + 3b + b2 = 0 + 12 3b + b2 = 0 + 12 Combine like terms: 0 + 12 = 12 3b + b2 = 12 The b term is 3b. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3b + 2.25 + b2 = 12 + 2.25 Reorder the terms: 2.25 + 3b + b2 = 12 + 2.25 Combine like terms: 12 + 2.25 = 14.25 2.25 + 3b + b2 = 14.25 Factor a perfect square on the left side: (b + 1.5)(b + 1.5) = 14.25 Calculate the square root of the right side: 3.774917218 Break this problem into two subproblems by setting (b + 1.5) equal to 3.774917218 and -3.774917218.Subproblem 1
b + 1.5 = 3.774917218 Simplifying b + 1.5 = 3.774917218 Reorder the terms: 1.5 + b = 3.774917218 Solving 1.5 + b = 3.774917218 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + b = 3.774917218 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + b = 3.774917218 + -1.5 b = 3.774917218 + -1.5 Combine like terms: 3.774917218 + -1.5 = 2.274917218 b = 2.274917218 Simplifying b = 2.274917218Subproblem 2
b + 1.5 = -3.774917218 Simplifying b + 1.5 = -3.774917218 Reorder the terms: 1.5 + b = -3.774917218 Solving 1.5 + b = -3.774917218 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + b = -3.774917218 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + b = -3.774917218 + -1.5 b = -3.774917218 + -1.5 Combine like terms: -3.774917218 + -1.5 = -5.274917218 b = -5.274917218 Simplifying b = -5.274917218Solution
The solution to the problem is based on the solutions from the subproblems. b = {2.274917218, -5.274917218}
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