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Simplifying (3b + -2)(b + 1) = 4 + -1(b + 2)(b + -1) Reorder the terms: (-2 + 3b)(b + 1) = 4 + -1(b + 2)(b + -1) Reorder the terms: (-2 + 3b)(1 + b) = 4 + -1(b + 2)(b + -1) Multiply (-2 + 3b) * (1 + b) (-2(1 + b) + 3b * (1 + b)) = 4 + -1(b + 2)(b + -1) ((1 * -2 + b * -2) + 3b * (1 + b)) = 4 + -1(b + 2)(b + -1) ((-2 + -2b) + 3b * (1 + b)) = 4 + -1(b + 2)(b + -1) (-2 + -2b + (1 * 3b + b * 3b)) = 4 + -1(b + 2)(b + -1) (-2 + -2b + (3b + 3b2)) = 4 + -1(b + 2)(b + -1) Combine like terms: -2b + 3b = 1b (-2 + 1b + 3b2) = 4 + -1(b + 2)(b + -1) Reorder the terms: -2 + 1b + 3b2 = 4 + -1(2 + b)(b + -1) Reorder the terms: -2 + 1b + 3b2 = 4 + -1(2 + b)(-1 + b) Multiply (2 + b) * (-1 + b) -2 + 1b + 3b2 = 4 + -1(2(-1 + b) + b(-1 + b)) -2 + 1b + 3b2 = 4 + -1((-1 * 2 + b * 2) + b(-1 + b)) -2 + 1b + 3b2 = 4 + -1((-2 + 2b) + b(-1 + b)) -2 + 1b + 3b2 = 4 + -1(-2 + 2b + (-1 * b + b * b)) -2 + 1b + 3b2 = 4 + -1(-2 + 2b + (-1b + b2)) Combine like terms: 2b + -1b = 1b -2 + 1b + 3b2 = 4 + -1(-2 + 1b + b2) -2 + 1b + 3b2 = 4 + (-2 * -1 + 1b * -1 + b2 * -1) -2 + 1b + 3b2 = 4 + (2 + -1b + -1b2) Combine like terms: 4 + 2 = 6 -2 + 1b + 3b2 = 6 + -1b + -1b2 Solving -2 + 1b + 3b2 = 6 + -1b + -1b2 Solving for variable 'b'. Reorder the terms: -2 + -6 + 1b + b + 3b2 + b2 = 6 + -1b + -1b2 + -6 + b + b2 Combine like terms: -2 + -6 = -8 -8 + 1b + b + 3b2 + b2 = 6 + -1b + -1b2 + -6 + b + b2 Combine like terms: 1b + b = 2b -8 + 2b + 3b2 + b2 = 6 + -1b + -1b2 + -6 + b + b2 Combine like terms: 3b2 + b2 = 4b2 -8 + 2b + 4b2 = 6 + -1b + -1b2 + -6 + b + b2 Reorder the terms: -8 + 2b + 4b2 = 6 + -6 + -1b + b + -1b2 + b2 Combine like terms: 6 + -6 = 0 -8 + 2b + 4b2 = 0 + -1b + b + -1b2 + b2 -8 + 2b + 4b2 = -1b + b + -1b2 + b2 Combine like terms: -1b + b = 0 -8 + 2b + 4b2 = 0 + -1b2 + b2 -8 + 2b + 4b2 = -1b2 + b2 Combine like terms: -1b2 + b2 = 0 -8 + 2b + 4b2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-4 + b + 2b2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-4 + b + 2b2)' equal to zero and attempt to solve: Simplifying -4 + b + 2b2 = 0 Solving -4 + b + 2b2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -2 + 0.5b + b2 = 0 Move the constant term to the right: Add '2' to each side of the equation. -2 + 0.5b + 2 + b2 = 0 + 2 Reorder the terms: -2 + 2 + 0.5b + b2 = 0 + 2 Combine like terms: -2 + 2 = 0 0 + 0.5b + b2 = 0 + 2 0.5b + b2 = 0 + 2 Combine like terms: 0 + 2 = 2 0.5b + b2 = 2 The b term is b. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. 0.5b + 0.25 + b2 = 2 + 0.25 Reorder the terms: 0.25 + 0.5b + b2 = 2 + 0.25 Combine like terms: 2 + 0.25 = 2.25 0.25 + 0.5b + b2 = 2.25 Factor a perfect square on the left side: (b + 0.5)(b + 0.5) = 2.25 Calculate the square root of the right side: 1.5 Break this problem into two subproblems by setting (b + 0.5) equal to 1.5 and -1.5.Subproblem 1
b + 0.5 = 1.5 Simplifying b + 0.5 = 1.5 Reorder the terms: 0.5 + b = 1.5 Solving 0.5 + b = 1.5 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + b = 1.5 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + b = 1.5 + -0.5 b = 1.5 + -0.5 Combine like terms: 1.5 + -0.5 = 1 b = 1 Simplifying b = 1Subproblem 2
b + 0.5 = -1.5 Simplifying b + 0.5 = -1.5 Reorder the terms: 0.5 + b = -1.5 Solving 0.5 + b = -1.5 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + b = -1.5 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + b = -1.5 + -0.5 b = -1.5 + -0.5 Combine like terms: -1.5 + -0.5 = -2 b = -2 Simplifying b = -2Solution
The solution to the problem is based on the solutions from the subproblems. b = {1, -2}Solution
b = {1, -2}
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