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Simplifying 6b2 + 3b2 + 8b + -9 + 24b2 + -14 + -5b + -4b = 0 Reorder the terms: -9 + -14 + 8b + -5b + -4b + 6b2 + 3b2 + 24b2 = 0 Combine like terms: -9 + -14 = -23 -23 + 8b + -5b + -4b + 6b2 + 3b2 + 24b2 = 0 Combine like terms: 8b + -5b = 3b -23 + 3b + -4b + 6b2 + 3b2 + 24b2 = 0 Combine like terms: 3b + -4b = -1b -23 + -1b + 6b2 + 3b2 + 24b2 = 0 Combine like terms: 6b2 + 3b2 = 9b2 -23 + -1b + 9b2 + 24b2 = 0 Combine like terms: 9b2 + 24b2 = 33b2 -23 + -1b + 33b2 = 0 Solving -23 + -1b + 33b2 = 0 Solving for variable 'b'. Begin completing the square. Divide all terms by 33 the coefficient of the squared term: Divide each side by '33'. -0.696969697 + -0.0303030303b + b2 = 0 Move the constant term to the right: Add '0.696969697' to each side of the equation. -0.696969697 + -0.0303030303b + 0.696969697 + b2 = 0 + 0.696969697 Reorder the terms: -0.696969697 + 0.696969697 + -0.0303030303b + b2 = 0 + 0.696969697 Combine like terms: -0.696969697 + 0.696969697 = 0.000000000 0.000000000 + -0.0303030303b + b2 = 0 + 0.696969697 -0.0303030303b + b2 = 0 + 0.696969697 Combine like terms: 0 + 0.696969697 = 0.696969697 -0.0303030303b + b2 = 0.696969697 The b term is -0.0303030303b. Take half its coefficient (-0.01515151515). Square it (0.0002295684113) and add it to both sides. Add '0.0002295684113' to each side of the equation. -0.0303030303b + 0.0002295684113 + b2 = 0.696969697 + 0.0002295684113 Reorder the terms: 0.0002295684113 + -0.0303030303b + b2 = 0.696969697 + 0.0002295684113 Combine like terms: 0.696969697 + 0.0002295684113 = 0.6971992654113 0.0002295684113 + -0.0303030303b + b2 = 0.6971992654113 Factor a perfect square on the left side: (b + -0.01515151515)(b + -0.01515151515) = 0.6971992654113 Calculate the square root of the right side: 0.83498459 Break this problem into two subproblems by setting (b + -0.01515151515) equal to 0.83498459 and -0.83498459.Subproblem 1
b + -0.01515151515 = 0.83498459 Simplifying b + -0.01515151515 = 0.83498459 Reorder the terms: -0.01515151515 + b = 0.83498459 Solving -0.01515151515 + b = 0.83498459 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '0.01515151515' to each side of the equation. -0.01515151515 + 0.01515151515 + b = 0.83498459 + 0.01515151515 Combine like terms: -0.01515151515 + 0.01515151515 = 0.00000000000 0.00000000000 + b = 0.83498459 + 0.01515151515 b = 0.83498459 + 0.01515151515 Combine like terms: 0.83498459 + 0.01515151515 = 0.85013610515 b = 0.85013610515 Simplifying b = 0.85013610515Subproblem 2
b + -0.01515151515 = -0.83498459 Simplifying b + -0.01515151515 = -0.83498459 Reorder the terms: -0.01515151515 + b = -0.83498459 Solving -0.01515151515 + b = -0.83498459 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '0.01515151515' to each side of the equation. -0.01515151515 + 0.01515151515 + b = -0.83498459 + 0.01515151515 Combine like terms: -0.01515151515 + 0.01515151515 = 0.00000000000 0.00000000000 + b = -0.83498459 + 0.01515151515 b = -0.83498459 + 0.01515151515 Combine like terms: -0.83498459 + 0.01515151515 = -0.81983307485 b = -0.81983307485 Simplifying b = -0.81983307485Solution
The solution to the problem is based on the solutions from the subproblems. b = {0.85013610515, -0.81983307485}
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