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Simplifying b2 + 3b + 5 = 35 Reorder the terms: 5 + 3b + b2 = 35 Solving 5 + 3b + b2 = 35 Solving for variable 'b'. Reorder the terms: 5 + -35 + 3b + b2 = 35 + -35 Combine like terms: 5 + -35 = -30 -30 + 3b + b2 = 35 + -35 Combine like terms: 35 + -35 = 0 -30 + 3b + b2 = 0 Begin completing the square. Move the constant term to the right: Add '30' to each side of the equation. -30 + 3b + 30 + b2 = 0 + 30 Reorder the terms: -30 + 30 + 3b + b2 = 0 + 30 Combine like terms: -30 + 30 = 0 0 + 3b + b2 = 0 + 30 3b + b2 = 0 + 30 Combine like terms: 0 + 30 = 30 3b + b2 = 30 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 = 30 + 2.25 Reorder the terms: 2.25 + 3b + b2 = 30 + 2.25 Combine like terms: 30 + 2.25 = 32.25 2.25 + 3b + b2 = 32.25 Factor a perfect square on the left side: (b + 1.5)(b + 1.5) = 32.25 Calculate the square root of the right side: 5.678908346 Break this problem into two subproblems by setting (b + 1.5) equal to 5.678908346 and -5.678908346.Subproblem 1
b + 1.5 = 5.678908346 Simplifying b + 1.5 = 5.678908346 Reorder the terms: 1.5 + b = 5.678908346 Solving 1.5 + b = 5.678908346 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 = 5.678908346 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + b = 5.678908346 + -1.5 b = 5.678908346 + -1.5 Combine like terms: 5.678908346 + -1.5 = 4.178908346 b = 4.178908346 Simplifying b = 4.178908346Subproblem 2
b + 1.5 = -5.678908346 Simplifying b + 1.5 = -5.678908346 Reorder the terms: 1.5 + b = -5.678908346 Solving 1.5 + b = -5.678908346 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 = -5.678908346 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + b = -5.678908346 + -1.5 b = -5.678908346 + -1.5 Combine like terms: -5.678908346 + -1.5 = -7.178908346 b = -7.178908346 Simplifying b = -7.178908346Solution
The solution to the problem is based on the solutions from the subproblems. b = {4.178908346, -7.178908346}
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