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Simplifying b2 + 6b(0.5) = 36 Reorder the terms for easier multiplication: b2 + 6 * 0.5b = 36 Multiply 6 * 0.5 b2 + 3b = 36 Reorder the terms: 3b + b2 = 36 Solving 3b + b2 = 36 Solving for variable 'b'. Reorder the terms: -36 + 3b + b2 = 36 + -36 Combine like terms: 36 + -36 = 0 -36 + 3b + b2 = 0 Begin completing the square. Move the constant term to the right: Add '36' to each side of the equation. -36 + 3b + 36 + b2 = 0 + 36 Reorder the terms: -36 + 36 + 3b + b2 = 0 + 36 Combine like terms: -36 + 36 = 0 0 + 3b + b2 = 0 + 36 3b + b2 = 0 + 36 Combine like terms: 0 + 36 = 36 3b + b2 = 36 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 = 36 + 2.25 Reorder the terms: 2.25 + 3b + b2 = 36 + 2.25 Combine like terms: 36 + 2.25 = 38.25 2.25 + 3b + b2 = 38.25 Factor a perfect square on the left side: (b + 1.5)(b + 1.5) = 38.25 Calculate the square root of the right side: 6.184658438 Break this problem into two subproblems by setting (b + 1.5) equal to 6.184658438 and -6.184658438.Subproblem 1
b + 1.5 = 6.184658438 Simplifying b + 1.5 = 6.184658438 Reorder the terms: 1.5 + b = 6.184658438 Solving 1.5 + b = 6.184658438 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 = 6.184658438 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + b = 6.184658438 + -1.5 b = 6.184658438 + -1.5 Combine like terms: 6.184658438 + -1.5 = 4.684658438 b = 4.684658438 Simplifying b = 4.684658438Subproblem 2
b + 1.5 = -6.184658438 Simplifying b + 1.5 = -6.184658438 Reorder the terms: 1.5 + b = -6.184658438 Solving 1.5 + b = -6.184658438 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 = -6.184658438 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + b = -6.184658438 + -1.5 b = -6.184658438 + -1.5 Combine like terms: -6.184658438 + -1.5 = -7.684658438 b = -7.684658438 Simplifying b = -7.684658438Solution
The solution to the problem is based on the solutions from the subproblems. b = {4.684658438, -7.684658438}
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