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Simplifying 6b2 + 5b + -27 = 0 Reorder the terms: -27 + 5b + 6b2 = 0 Solving -27 + 5b + 6b2 = 0 Solving for variable 'b'. Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -4.5 + 0.8333333333b + b2 = 0 Move the constant term to the right: Add '4.5' to each side of the equation. -4.5 + 0.8333333333b + 4.5 + b2 = 0 + 4.5 Reorder the terms: -4.5 + 4.5 + 0.8333333333b + b2 = 0 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + 0.8333333333b + b2 = 0 + 4.5 0.8333333333b + b2 = 0 + 4.5 Combine like terms: 0 + 4.5 = 4.5 0.8333333333b + b2 = 4.5 The b term is 0.8333333333b. Take half its coefficient (0.4166666667). Square it (0.1736111111) and add it to both sides. Add '0.1736111111' to each side of the equation. 0.8333333333b + 0.1736111111 + b2 = 4.5 + 0.1736111111 Reorder the terms: 0.1736111111 + 0.8333333333b + b2 = 4.5 + 0.1736111111 Combine like terms: 4.5 + 0.1736111111 = 4.6736111111 0.1736111111 + 0.8333333333b + b2 = 4.6736111111 Factor a perfect square on the left side: (b + 0.4166666667)(b + 0.4166666667) = 4.6736111111 Calculate the square root of the right side: 2.161853629 Break this problem into two subproblems by setting (b + 0.4166666667) equal to 2.161853629 and -2.161853629.Subproblem 1
b + 0.4166666667 = 2.161853629 Simplifying b + 0.4166666667 = 2.161853629 Reorder the terms: 0.4166666667 + b = 2.161853629 Solving 0.4166666667 + b = 2.161853629 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-0.4166666667' to each side of the equation. 0.4166666667 + -0.4166666667 + b = 2.161853629 + -0.4166666667 Combine like terms: 0.4166666667 + -0.4166666667 = 0.0000000000 0.0000000000 + b = 2.161853629 + -0.4166666667 b = 2.161853629 + -0.4166666667 Combine like terms: 2.161853629 + -0.4166666667 = 1.7451869623 b = 1.7451869623 Simplifying b = 1.7451869623Subproblem 2
b + 0.4166666667 = -2.161853629 Simplifying b + 0.4166666667 = -2.161853629 Reorder the terms: 0.4166666667 + b = -2.161853629 Solving 0.4166666667 + b = -2.161853629 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-0.4166666667' to each side of the equation. 0.4166666667 + -0.4166666667 + b = -2.161853629 + -0.4166666667 Combine like terms: 0.4166666667 + -0.4166666667 = 0.0000000000 0.0000000000 + b = -2.161853629 + -0.4166666667 b = -2.161853629 + -0.4166666667 Combine like terms: -2.161853629 + -0.4166666667 = -2.5785202957 b = -2.5785202957 Simplifying b = -2.5785202957Solution
The solution to the problem is based on the solutions from the subproblems. b = {1.7451869623, -2.5785202957}
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