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Simplifying b(6b + 7) = 8 Reorder the terms: b(7 + 6b) = 8 (7 * b + 6b * b) = 8 (7b + 6b2) = 8 Solving 7b + 6b2 = 8 Solving for variable 'b'. Reorder the terms: -8 + 7b + 6b2 = 8 + -8 Combine like terms: 8 + -8 = 0 -8 + 7b + 6b2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -1.333333333 + 1.166666667b + b2 = 0 Move the constant term to the right: Add '1.333333333' to each side of the equation. -1.333333333 + 1.166666667b + 1.333333333 + b2 = 0 + 1.333333333 Reorder the terms: -1.333333333 + 1.333333333 + 1.166666667b + b2 = 0 + 1.333333333 Combine like terms: -1.333333333 + 1.333333333 = 0.000000000 0.000000000 + 1.166666667b + b2 = 0 + 1.333333333 1.166666667b + b2 = 0 + 1.333333333 Combine like terms: 0 + 1.333333333 = 1.333333333 1.166666667b + b2 = 1.333333333 The b term is 1.166666667b. Take half its coefficient (0.5833333335). Square it (0.3402777780) and add it to both sides. Add '0.3402777780' to each side of the equation. 1.166666667b + 0.3402777780 + b2 = 1.333333333 + 0.3402777780 Reorder the terms: 0.3402777780 + 1.166666667b + b2 = 1.333333333 + 0.3402777780 Combine like terms: 1.333333333 + 0.3402777780 = 1.673611111 0.3402777780 + 1.166666667b + b2 = 1.673611111 Factor a perfect square on the left side: (b + 0.5833333335)(b + 0.5833333335) = 1.673611111 Calculate the square root of the right side: 1.293681225 Break this problem into two subproblems by setting (b + 0.5833333335) equal to 1.293681225 and -1.293681225.Subproblem 1
b + 0.5833333335 = 1.293681225 Simplifying b + 0.5833333335 = 1.293681225 Reorder the terms: 0.5833333335 + b = 1.293681225 Solving 0.5833333335 + b = 1.293681225 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-0.5833333335' to each side of the equation. 0.5833333335 + -0.5833333335 + b = 1.293681225 + -0.5833333335 Combine like terms: 0.5833333335 + -0.5833333335 = 0.0000000000 0.0000000000 + b = 1.293681225 + -0.5833333335 b = 1.293681225 + -0.5833333335 Combine like terms: 1.293681225 + -0.5833333335 = 0.7103478915 b = 0.7103478915 Simplifying b = 0.7103478915Subproblem 2
b + 0.5833333335 = -1.293681225 Simplifying b + 0.5833333335 = -1.293681225 Reorder the terms: 0.5833333335 + b = -1.293681225 Solving 0.5833333335 + b = -1.293681225 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-0.5833333335' to each side of the equation. 0.5833333335 + -0.5833333335 + b = -1.293681225 + -0.5833333335 Combine like terms: 0.5833333335 + -0.5833333335 = 0.0000000000 0.0000000000 + b = -1.293681225 + -0.5833333335 b = -1.293681225 + -0.5833333335 Combine like terms: -1.293681225 + -0.5833333335 = -1.8770145585 b = -1.8770145585 Simplifying b = -1.8770145585Solution
The solution to the problem is based on the solutions from the subproblems. b = {0.7103478915, -1.8770145585}
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