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Simplifying (3.2 + -1b) * b = 2.56 Reorder the terms for easier multiplication: b(3.2 + -1b) = 2.56 (3.2 * b + -1b * b) = 2.56 (3.2b + -1b^{2}) = 2.56 Solving 3.2b + -1b^{2}= 2.56 Solving for variable 'b'. Reorder the terms: -2.56 + 3.2b + -1b^{2}= 2.56 + -2.56 Combine like terms: 2.56 + -2.56 = 0.00 -2.56 + 3.2b + -1b^{2}= 0.00 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. 2.56 + -3.2b + b^{2}= 0.00 Move the constant term to the right: Add '-2.56' to each side of the equation. 2.56 + -3.2b + -2.56 + b^{2}= 0.00 + -2.56 Reorder the terms: 2.56 + -2.56 + -3.2b + b^{2}= 0.00 + -2.56 Combine like terms: 2.56 + -2.56 = 0.00 0.00 + -3.2b + b^{2}= 0.00 + -2.56 -3.2b + b^{2}= 0.00 + -2.56 Combine like terms: 0.00 + -2.56 = -2.56 -3.2b + b^{2}= -2.56 The b term is -3.2b. Take half its coefficient (-1.6). Square it (2.56) and add it to both sides. Add '2.56' to each side of the equation. -3.2b + 2.56 + b^{2}= -2.56 + 2.56 Reorder the terms: 2.56 + -3.2b + b^{2}= -2.56 + 2.56 Combine like terms: -2.56 + 2.56 = 0.00 2.56 + -3.2b + b^{2}= 0.00 Factor a perfect square on the left side: (b + -1.6)(b + -1.6) = 0.00 Calculate the square root of the right side: 0.00 Break this problem into two subproblems by setting (b + -1.6) equal to 0.00 and 0.00.## Subproblem 1

b + -1.6 = 0.00 Simplifying b + -1.6 = 0.00 Reorder the terms: -1.6 + b = 0.00 Solving -1.6 + b = 0.00 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '1.6' to each side of the equation. -1.6 + 1.6 + b = 0.00 + 1.6 Combine like terms: -1.6 + 1.6 = 0.0 0.0 + b = 0.00 + 1.6 b = 0.00 + 1.6 Combine like terms: 0.00 + 1.6 = 1.6 b = 1.6 Simplifying b = 1.6## Subproblem 2

b + -1.6 = 0.00 Simplifying b + -1.6 = 0.00 Reorder the terms: -1.6 + b = 0.00 Solving -1.6 + b = 0.00 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '1.6' to each side of the equation. -1.6 + 1.6 + b = 0.00 + 1.6 Combine like terms: -1.6 + 1.6 = 0.0 0.0 + b = 0.00 + 1.6 b = 0.00 + 1.6 Combine like terms: 0.00 + 1.6 = 1.6 b = 1.6 Simplifying b = 1.6## Solution

The solution to the problem is based on the solutions from the subproblems. b = {1.6, 1.6}

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