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Simplifying 30b2 + 33b + -56 = 0 Reorder the terms: -56 + 33b + 30b2 = 0 Solving -56 + 33b + 30b2 = 0 Solving for variable 'b'. Begin completing the square. Divide all terms by 30 the coefficient of the squared term: Divide each side by '30'. -1.866666667 + 1.1b + b2 = 0 Move the constant term to the right: Add '1.866666667' to each side of the equation. -1.866666667 + 1.1b + 1.866666667 + b2 = 0 + 1.866666667 Reorder the terms: -1.866666667 + 1.866666667 + 1.1b + b2 = 0 + 1.866666667 Combine like terms: -1.866666667 + 1.866666667 = 0.000000000 0.000000000 + 1.1b + b2 = 0 + 1.866666667 1.1b + b2 = 0 + 1.866666667 Combine like terms: 0 + 1.866666667 = 1.866666667 1.1b + b2 = 1.866666667 The b term is 1.1b. Take half its coefficient (0.55). Square it (0.3025) and add it to both sides. Add '0.3025' to each side of the equation. 1.1b + 0.3025 + b2 = 1.866666667 + 0.3025 Reorder the terms: 0.3025 + 1.1b + b2 = 1.866666667 + 0.3025 Combine like terms: 1.866666667 + 0.3025 = 2.169166667 0.3025 + 1.1b + b2 = 2.169166667 Factor a perfect square on the left side: (b + 0.55)(b + 0.55) = 2.169166667 Calculate the square root of the right side: 1.472809107 Break this problem into two subproblems by setting (b + 0.55) equal to 1.472809107 and -1.472809107.Subproblem 1
b + 0.55 = 1.472809107 Simplifying b + 0.55 = 1.472809107 Reorder the terms: 0.55 + b = 1.472809107 Solving 0.55 + b = 1.472809107 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-0.55' to each side of the equation. 0.55 + -0.55 + b = 1.472809107 + -0.55 Combine like terms: 0.55 + -0.55 = 0.00 0.00 + b = 1.472809107 + -0.55 b = 1.472809107 + -0.55 Combine like terms: 1.472809107 + -0.55 = 0.922809107 b = 0.922809107 Simplifying b = 0.922809107Subproblem 2
b + 0.55 = -1.472809107 Simplifying b + 0.55 = -1.472809107 Reorder the terms: 0.55 + b = -1.472809107 Solving 0.55 + b = -1.472809107 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-0.55' to each side of the equation. 0.55 + -0.55 + b = -1.472809107 + -0.55 Combine like terms: 0.55 + -0.55 = 0.00 0.00 + b = -1.472809107 + -0.55 b = -1.472809107 + -0.55 Combine like terms: -1.472809107 + -0.55 = -2.022809107 b = -2.022809107 Simplifying b = -2.022809107Solution
The solution to the problem is based on the solutions from the subproblems. b = {0.922809107, -2.022809107}
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