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Simplifying (3b + -4)(6b + 5) + -2 = 0 Reorder the terms: (-4 + 3b)(6b + 5) + -2 = 0 Reorder the terms: (-4 + 3b)(5 + 6b) + -2 = 0 Multiply (-4 + 3b) * (5 + 6b) (-4(5 + 6b) + 3b * (5 + 6b)) + -2 = 0 ((5 * -4 + 6b * -4) + 3b * (5 + 6b)) + -2 = 0 ((-20 + -24b) + 3b * (5 + 6b)) + -2 = 0 (-20 + -24b + (5 * 3b + 6b * 3b)) + -2 = 0 (-20 + -24b + (15b + 18b2)) + -2 = 0 Combine like terms: -24b + 15b = -9b (-20 + -9b + 18b2) + -2 = 0 Reorder the terms: -20 + -2 + -9b + 18b2 = 0 Combine like terms: -20 + -2 = -22 -22 + -9b + 18b2 = 0 Solving -22 + -9b + 18b2 = 0 Solving for variable 'b'. Begin completing the square. Divide all terms by 18 the coefficient of the squared term: Divide each side by '18'. -1.222222222 + -0.5b + b2 = 0 Move the constant term to the right: Add '1.222222222' to each side of the equation. -1.222222222 + -0.5b + 1.222222222 + b2 = 0 + 1.222222222 Reorder the terms: -1.222222222 + 1.222222222 + -0.5b + b2 = 0 + 1.222222222 Combine like terms: -1.222222222 + 1.222222222 = 0.000000000 0.000000000 + -0.5b + b2 = 0 + 1.222222222 -0.5b + b2 = 0 + 1.222222222 Combine like terms: 0 + 1.222222222 = 1.222222222 -0.5b + b2 = 1.222222222 The b term is -0.5b. Take half its coefficient (-0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. -0.5b + 0.0625 + b2 = 1.222222222 + 0.0625 Reorder the terms: 0.0625 + -0.5b + b2 = 1.222222222 + 0.0625 Combine like terms: 1.222222222 + 0.0625 = 1.284722222 0.0625 + -0.5b + b2 = 1.284722222 Factor a perfect square on the left side: (b + -0.25)(b + -0.25) = 1.284722222 Calculate the square root of the right side: 1.133455876 Break this problem into two subproblems by setting (b + -0.25) equal to 1.133455876 and -1.133455876.Subproblem 1
b + -0.25 = 1.133455876 Simplifying b + -0.25 = 1.133455876 Reorder the terms: -0.25 + b = 1.133455876 Solving -0.25 + b = 1.133455876 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + b = 1.133455876 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + b = 1.133455876 + 0.25 b = 1.133455876 + 0.25 Combine like terms: 1.133455876 + 0.25 = 1.383455876 b = 1.383455876 Simplifying b = 1.383455876Subproblem 2
b + -0.25 = -1.133455876 Simplifying b + -0.25 = -1.133455876 Reorder the terms: -0.25 + b = -1.133455876 Solving -0.25 + b = -1.133455876 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + b = -1.133455876 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + b = -1.133455876 + 0.25 b = -1.133455876 + 0.25 Combine like terms: -1.133455876 + 0.25 = -0.883455876 b = -0.883455876 Simplifying b = -0.883455876Solution
The solution to the problem is based on the solutions from the subproblems. b = {1.383455876, -0.883455876}
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