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Simplifying b2 = b + 32 Reorder the terms: b2 = 32 + b Solving b2 = 32 + b Solving for variable 'b'. Reorder the terms: -32 + -1b + b2 = 32 + b + -32 + -1b Reorder the terms: -32 + -1b + b2 = 32 + -32 + b + -1b Combine like terms: 32 + -32 = 0 -32 + -1b + b2 = 0 + b + -1b -32 + -1b + b2 = b + -1b Combine like terms: b + -1b = 0 -32 + -1b + b2 = 0 Begin completing the square. Move the constant term to the right: Add '32' to each side of the equation. -32 + -1b + 32 + b2 = 0 + 32 Reorder the terms: -32 + 32 + -1b + b2 = 0 + 32 Combine like terms: -32 + 32 = 0 0 + -1b + b2 = 0 + 32 -1b + b2 = 0 + 32 Combine like terms: 0 + 32 = 32 -1b + b2 = 32 The b term is -1b. 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. -1b + 0.25 + b2 = 32 + 0.25 Reorder the terms: 0.25 + -1b + b2 = 32 + 0.25 Combine like terms: 32 + 0.25 = 32.25 0.25 + -1b + b2 = 32.25 Factor a perfect square on the left side: (b + -0.5)(b + -0.5) = 32.25 Calculate the square root of the right side: 5.678908346 Break this problem into two subproblems by setting (b + -0.5) equal to 5.678908346 and -5.678908346.Subproblem 1
b + -0.5 = 5.678908346 Simplifying b + -0.5 = 5.678908346 Reorder the terms: -0.5 + b = 5.678908346 Solving -0.5 + b = 5.678908346 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 = 5.678908346 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + b = 5.678908346 + 0.5 b = 5.678908346 + 0.5 Combine like terms: 5.678908346 + 0.5 = 6.178908346 b = 6.178908346 Simplifying b = 6.178908346Subproblem 2
b + -0.5 = -5.678908346 Simplifying b + -0.5 = -5.678908346 Reorder the terms: -0.5 + b = -5.678908346 Solving -0.5 + b = -5.678908346 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 = -5.678908346 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + b = -5.678908346 + 0.5 b = -5.678908346 + 0.5 Combine like terms: -5.678908346 + 0.5 = -5.178908346 b = -5.178908346 Simplifying b = -5.178908346Solution
The solution to the problem is based on the solutions from the subproblems. b = {6.178908346, -5.178908346}
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