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Simplifying b(14b + 3) = 10 Reorder the terms: b(3 + 14b) = 10 (3 * b + 14b * b) = 10 (3b + 14b2) = 10 Solving 3b + 14b2 = 10 Solving for variable 'b'. Reorder the terms: -10 + 3b + 14b2 = 10 + -10 Combine like terms: 10 + -10 = 0 -10 + 3b + 14b2 = 0 Begin completing the square. Divide all terms by 14 the coefficient of the squared term: Divide each side by '14'. -0.7142857143 + 0.2142857143b + b2 = 0 Move the constant term to the right: Add '0.7142857143' to each side of the equation. -0.7142857143 + 0.2142857143b + 0.7142857143 + b2 = 0 + 0.7142857143 Reorder the terms: -0.7142857143 + 0.7142857143 + 0.2142857143b + b2 = 0 + 0.7142857143 Combine like terms: -0.7142857143 + 0.7142857143 = 0.0000000000 0.0000000000 + 0.2142857143b + b2 = 0 + 0.7142857143 0.2142857143b + b2 = 0 + 0.7142857143 Combine like terms: 0 + 0.7142857143 = 0.7142857143 0.2142857143b + b2 = 0.7142857143 The b term is 0.2142857143b. Take half its coefficient (0.1071428572). Square it (0.01147959185) and add it to both sides. Add '0.01147959185' to each side of the equation. 0.2142857143b + 0.01147959185 + b2 = 0.7142857143 + 0.01147959185 Reorder the terms: 0.01147959185 + 0.2142857143b + b2 = 0.7142857143 + 0.01147959185 Combine like terms: 0.7142857143 + 0.01147959185 = 0.72576530615 0.01147959185 + 0.2142857143b + b2 = 0.72576530615 Factor a perfect square on the left side: (b + 0.1071428572)(b + 0.1071428572) = 0.72576530615 Calculate the square root of the right side: 0.851918603 Break this problem into two subproblems by setting (b + 0.1071428572) equal to 0.851918603 and -0.851918603.Subproblem 1
b + 0.1071428572 = 0.851918603 Simplifying b + 0.1071428572 = 0.851918603 Reorder the terms: 0.1071428572 + b = 0.851918603 Solving 0.1071428572 + b = 0.851918603 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-0.1071428572' to each side of the equation. 0.1071428572 + -0.1071428572 + b = 0.851918603 + -0.1071428572 Combine like terms: 0.1071428572 + -0.1071428572 = 0.0000000000 0.0000000000 + b = 0.851918603 + -0.1071428572 b = 0.851918603 + -0.1071428572 Combine like terms: 0.851918603 + -0.1071428572 = 0.7447757458 b = 0.7447757458 Simplifying b = 0.7447757458Subproblem 2
b + 0.1071428572 = -0.851918603 Simplifying b + 0.1071428572 = -0.851918603 Reorder the terms: 0.1071428572 + b = -0.851918603 Solving 0.1071428572 + b = -0.851918603 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-0.1071428572' to each side of the equation. 0.1071428572 + -0.1071428572 + b = -0.851918603 + -0.1071428572 Combine like terms: 0.1071428572 + -0.1071428572 = 0.0000000000 0.0000000000 + b = -0.851918603 + -0.1071428572 b = -0.851918603 + -0.1071428572 Combine like terms: -0.851918603 + -0.1071428572 = -0.9590614602 b = -0.9590614602 Simplifying b = -0.9590614602Solution
The solution to the problem is based on the solutions from the subproblems. b = {0.7447757458, -0.9590614602}
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