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Simplifying 6b2 + 18b + 11 = 0 Reorder the terms: 11 + 18b + 6b2 = 0 Solving 11 + 18b + 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'. 1.833333333 + 3b + b2 = 0 Move the constant term to the right: Add '-1.833333333' to each side of the equation. 1.833333333 + 3b + -1.833333333 + b2 = 0 + -1.833333333 Reorder the terms: 1.833333333 + -1.833333333 + 3b + b2 = 0 + -1.833333333 Combine like terms: 1.833333333 + -1.833333333 = 0.000000000 0.000000000 + 3b + b2 = 0 + -1.833333333 3b + b2 = 0 + -1.833333333 Combine like terms: 0 + -1.833333333 = -1.833333333 3b + b2 = -1.833333333 The b term is 3b. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3b + 2.25 + b2 = -1.833333333 + 2.25 Reorder the terms: 2.25 + 3b + b2 = -1.833333333 + 2.25 Combine like terms: -1.833333333 + 2.25 = 0.416666667 2.25 + 3b + b2 = 0.416666667 Factor a perfect square on the left side: (b + 1.5)(b + 1.5) = 0.416666667 Calculate the square root of the right side: 0.645497225 Break this problem into two subproblems by setting (b + 1.5) equal to 0.645497225 and -0.645497225.Subproblem 1
b + 1.5 = 0.645497225 Simplifying b + 1.5 = 0.645497225 Reorder the terms: 1.5 + b = 0.645497225 Solving 1.5 + b = 0.645497225 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + b = 0.645497225 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + b = 0.645497225 + -1.5 b = 0.645497225 + -1.5 Combine like terms: 0.645497225 + -1.5 = -0.854502775 b = -0.854502775 Simplifying b = -0.854502775Subproblem 2
b + 1.5 = -0.645497225 Simplifying b + 1.5 = -0.645497225 Reorder the terms: 1.5 + b = -0.645497225 Solving 1.5 + b = -0.645497225 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + b = -0.645497225 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + b = -0.645497225 + -1.5 b = -0.645497225 + -1.5 Combine like terms: -0.645497225 + -1.5 = -2.145497225 b = -2.145497225 Simplifying b = -2.145497225Solution
The solution to the problem is based on the solutions from the subproblems. b = {-0.854502775, -2.145497225}
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