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Simplifying 2b + b(b + 16) = 180 Reorder the terms: 2b + b(16 + b) = 180 2b + (16 * b + b * b) = 180 2b + (16b + b2) = 180 Combine like terms: 2b + 16b = 18b 18b + b2 = 180 Solving 18b + b2 = 180 Solving for variable 'b'. Reorder the terms: -180 + 18b + b2 = 180 + -180 Combine like terms: 180 + -180 = 0 -180 + 18b + b2 = 0 Begin completing the square. Move the constant term to the right: Add '180' to each side of the equation. -180 + 18b + 180 + b2 = 0 + 180 Reorder the terms: -180 + 180 + 18b + b2 = 0 + 180 Combine like terms: -180 + 180 = 0 0 + 18b + b2 = 0 + 180 18b + b2 = 0 + 180 Combine like terms: 0 + 180 = 180 18b + b2 = 180 The b term is 18b. Take half its coefficient (9). Square it (81) and add it to both sides. Add '81' to each side of the equation. 18b + 81 + b2 = 180 + 81 Reorder the terms: 81 + 18b + b2 = 180 + 81 Combine like terms: 180 + 81 = 261 81 + 18b + b2 = 261 Factor a perfect square on the left side: (b + 9)(b + 9) = 261 Calculate the square root of the right side: 16.155494421 Break this problem into two subproblems by setting (b + 9) equal to 16.155494421 and -16.155494421.Subproblem 1
b + 9 = 16.155494421 Simplifying b + 9 = 16.155494421 Reorder the terms: 9 + b = 16.155494421 Solving 9 + b = 16.155494421 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-9' to each side of the equation. 9 + -9 + b = 16.155494421 + -9 Combine like terms: 9 + -9 = 0 0 + b = 16.155494421 + -9 b = 16.155494421 + -9 Combine like terms: 16.155494421 + -9 = 7.155494421 b = 7.155494421 Simplifying b = 7.155494421Subproblem 2
b + 9 = -16.155494421 Simplifying b + 9 = -16.155494421 Reorder the terms: 9 + b = -16.155494421 Solving 9 + b = -16.155494421 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-9' to each side of the equation. 9 + -9 + b = -16.155494421 + -9 Combine like terms: 9 + -9 = 0 0 + b = -16.155494421 + -9 b = -16.155494421 + -9 Combine like terms: -16.155494421 + -9 = -25.155494421 b = -25.155494421 Simplifying b = -25.155494421Solution
The solution to the problem is based on the solutions from the subproblems. b = {7.155494421, -25.155494421}
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