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Simplifying (b + 0.5)(b + 0.5) = 29.75 Reorder the terms: (0.5 + b)(b + 0.5) = 29.75 Reorder the terms: (0.5 + b)(0.5 + b) = 29.75 Multiply (0.5 + b) * (0.5 + b) (0.5(0.5 + b) + b(0.5 + b)) = 29.75 ((0.5 * 0.5 + b * 0.5) + b(0.5 + b)) = 29.75 ((0.25 + 0.5b) + b(0.5 + b)) = 29.75 (0.25 + 0.5b + (0.5 * b + b * b)) = 29.75 (0.25 + 0.5b + (0.5b + b2)) = 29.75 Combine like terms: 0.5b + 0.5b = 1b (0.25 + 1b + b2) = 29.75 Solving 0.25 + 1b + b2 = 29.75 Solving for variable 'b'. Reorder the terms: 0.25 + -29.75 + 1b + b2 = 29.75 + -29.75 Combine like terms: 0.25 + -29.75 = -29.5 -29.5 + 1b + b2 = 29.75 + -29.75 Combine like terms: 29.75 + -29.75 = 0.00 -29.5 + 1b + b2 = 0.00 Begin completing the square. Move the constant term to the right: Add '29.5' to each side of the equation. -29.5 + 1b + 29.5 + b2 = 0.00 + 29.5 Reorder the terms: -29.5 + 29.5 + 1b + b2 = 0.00 + 29.5 Combine like terms: -29.5 + 29.5 = 0.0 0.0 + 1b + b2 = 0.00 + 29.5 1b + b2 = 0.00 + 29.5 Combine like terms: 0.00 + 29.5 = 29.5 1b + b2 = 29.5 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. b + 0.25 + b2 = 29.5 + 0.25 Reorder the terms: 0.25 + b + b2 = 29.5 + 0.25 Combine like terms: 29.5 + 0.25 = 29.75 0.25 + b + b2 = 29.75 Factor a perfect square on the left side: (b + 0.5)(b + 0.5) = 29.75 Calculate the square root of the right side: 5.454356057 Break this problem into two subproblems by setting (b + 0.5) equal to 5.454356057 and -5.454356057.Subproblem 1
b + 0.5 = 5.454356057 Simplifying b + 0.5 = 5.454356057 Reorder the terms: 0.5 + b = 5.454356057 Solving 0.5 + b = 5.454356057 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 = 5.454356057 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + b = 5.454356057 + -0.5 b = 5.454356057 + -0.5 Combine like terms: 5.454356057 + -0.5 = 4.954356057 b = 4.954356057 Simplifying b = 4.954356057Subproblem 2
b + 0.5 = -5.454356057 Simplifying b + 0.5 = -5.454356057 Reorder the terms: 0.5 + b = -5.454356057 Solving 0.5 + b = -5.454356057 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 = -5.454356057 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + b = -5.454356057 + -0.5 b = -5.454356057 + -0.5 Combine like terms: -5.454356057 + -0.5 = -5.954356057 b = -5.954356057 Simplifying b = -5.954356057Solution
The solution to the problem is based on the solutions from the subproblems. b = {4.954356057, -5.954356057}
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