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Simplifying (b + -3)(3b + -1) + -2b(b + 2) = 0 Reorder the terms: (-3 + b)(3b + -1) + -2b(b + 2) = 0 Reorder the terms: (-3 + b)(-1 + 3b) + -2b(b + 2) = 0 Multiply (-3 + b) * (-1 + 3b) (-3(-1 + 3b) + b(-1 + 3b)) + -2b(b + 2) = 0 ((-1 * -3 + 3b * -3) + b(-1 + 3b)) + -2b(b + 2) = 0 ((3 + -9b) + b(-1 + 3b)) + -2b(b + 2) = 0 (3 + -9b + (-1 * b + 3b * b)) + -2b(b + 2) = 0 (3 + -9b + (-1b + 3b2)) + -2b(b + 2) = 0 Combine like terms: -9b + -1b = -10b (3 + -10b + 3b2) + -2b(b + 2) = 0 Reorder the terms: 3 + -10b + 3b2 + -2b(2 + b) = 0 3 + -10b + 3b2 + (2 * -2b + b * -2b) = 0 3 + -10b + 3b2 + (-4b + -2b2) = 0 Reorder the terms: 3 + -10b + -4b + 3b2 + -2b2 = 0 Combine like terms: -10b + -4b = -14b 3 + -14b + 3b2 + -2b2 = 0 Combine like terms: 3b2 + -2b2 = 1b2 3 + -14b + 1b2 = 0 Solving 3 + -14b + 1b2 = 0 Solving for variable 'b'. Begin completing the square. Move the constant term to the right: Add '-3' to each side of the equation. 3 + -14b + -3 + b2 = 0 + -3 Reorder the terms: 3 + -3 + -14b + b2 = 0 + -3 Combine like terms: 3 + -3 = 0 0 + -14b + b2 = 0 + -3 -14b + b2 = 0 + -3 Combine like terms: 0 + -3 = -3 -14b + b2 = -3 The b term is -14b. Take half its coefficient (-7). Square it (49) and add it to both sides. Add '49' to each side of the equation. -14b + 49 + b2 = -3 + 49 Reorder the terms: 49 + -14b + b2 = -3 + 49 Combine like terms: -3 + 49 = 46 49 + -14b + b2 = 46 Factor a perfect square on the left side: (b + -7)(b + -7) = 46 Calculate the square root of the right side: 6.782329983 Break this problem into two subproblems by setting (b + -7) equal to 6.782329983 and -6.782329983.Subproblem 1
b + -7 = 6.782329983 Simplifying b + -7 = 6.782329983 Reorder the terms: -7 + b = 6.782329983 Solving -7 + b = 6.782329983 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '7' to each side of the equation. -7 + 7 + b = 6.782329983 + 7 Combine like terms: -7 + 7 = 0 0 + b = 6.782329983 + 7 b = 6.782329983 + 7 Combine like terms: 6.782329983 + 7 = 13.782329983 b = 13.782329983 Simplifying b = 13.782329983Subproblem 2
b + -7 = -6.782329983 Simplifying b + -7 = -6.782329983 Reorder the terms: -7 + b = -6.782329983 Solving -7 + b = -6.782329983 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '7' to each side of the equation. -7 + 7 + b = -6.782329983 + 7 Combine like terms: -7 + 7 = 0 0 + b = -6.782329983 + 7 b = -6.782329983 + 7 Combine like terms: -6.782329983 + 7 = 0.217670017 b = 0.217670017 Simplifying b = 0.217670017Solution
The solution to the problem is based on the solutions from the subproblems. b = {13.782329983, 0.217670017}
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