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Simplifying (1 + -1b) * b = 0.09 Reorder the terms for easier multiplication: b(1 + -1b) = 0.09 (1 * b + -1b * b) = 0.09 (1b + -1b2) = 0.09 Solving 1b + -1b2 = 0.09 Solving for variable 'b'. Reorder the terms: -0.09 + 1b + -1b2 = 0.09 + -0.09 Combine like terms: 0.09 + -0.09 = 0.00 -0.09 + 1b + -1b2 = 0.00 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. 0.09 + -1b + b2 = 0.00 Move the constant term to the right: Add '-0.09' to each side of the equation. 0.09 + -1b + -0.09 + b2 = 0.00 + -0.09 Reorder the terms: 0.09 + -0.09 + -1b + b2 = 0.00 + -0.09 Combine like terms: 0.09 + -0.09 = 0.00 0.00 + -1b + b2 = 0.00 + -0.09 -1b + b2 = 0.00 + -0.09 Combine like terms: 0.00 + -0.09 = -0.09 -1b + b2 = -0.09 The b term is -1b. 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. -1b + 0.25 + b2 = -0.09 + 0.25 Reorder the terms: 0.25 + -1b + b2 = -0.09 + 0.25 Combine like terms: -0.09 + 0.25 = 0.16 0.25 + -1b + b2 = 0.16 Factor a perfect square on the left side: (b + -0.5)(b + -0.5) = 0.16 Calculate the square root of the right side: 0.4 Break this problem into two subproblems by setting (b + -0.5) equal to 0.4 and -0.4.Subproblem 1
b + -0.5 = 0.4 Simplifying b + -0.5 = 0.4 Reorder the terms: -0.5 + b = 0.4 Solving -0.5 + b = 0.4 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 = 0.4 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + b = 0.4 + 0.5 b = 0.4 + 0.5 Combine like terms: 0.4 + 0.5 = 0.9 b = 0.9 Simplifying b = 0.9Subproblem 2
b + -0.5 = -0.4 Simplifying b + -0.5 = -0.4 Reorder the terms: -0.5 + b = -0.4 Solving -0.5 + b = -0.4 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 = -0.4 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + b = -0.4 + 0.5 b = -0.4 + 0.5 Combine like terms: -0.4 + 0.5 = 0.1 b = 0.1 Simplifying b = 0.1Solution
The solution to the problem is based on the solutions from the subproblems. b = {0.9, 0.1}
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