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Simplifying (0.5b + 0.4)(b + 0.7) = 0 Reorder the terms: (0.4 + 0.5b)(b + 0.7) = 0 Reorder the terms: (0.4 + 0.5b)(0.7 + b) = 0 Multiply (0.4 + 0.5b) * (0.7 + b) (0.4(0.7 + b) + 0.5b * (0.7 + b)) = 0 ((0.7 * 0.4 + b * 0.4) + 0.5b * (0.7 + b)) = 0 ((0.28 + 0.4b) + 0.5b * (0.7 + b)) = 0 (0.28 + 0.4b + (0.7 * 0.5b + b * 0.5b)) = 0 (0.28 + 0.4b + (0.35b + 0.5b2)) = 0 Combine like terms: 0.4b + 0.35b = 0.75b (0.28 + 0.75b + 0.5b2) = 0 Solving 0.28 + 0.75b + 0.5b2 = 0 Solving for variable 'b'. Begin completing the square. Divide all terms by 0.5 the coefficient of the squared term: Divide each side by '0.5'. 0.56 + 1.5b + b2 = 0 Move the constant term to the right: Add '-0.56' to each side of the equation. 0.56 + 1.5b + -0.56 + b2 = 0 + -0.56 Reorder the terms: 0.56 + -0.56 + 1.5b + b2 = 0 + -0.56 Combine like terms: 0.56 + -0.56 = 0.00 0.00 + 1.5b + b2 = 0 + -0.56 1.5b + b2 = 0 + -0.56 Combine like terms: 0 + -0.56 = -0.56 1.5b + b2 = -0.56 The b term is 1.5b. Take half its coefficient (0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. 1.5b + 0.5625 + b2 = -0.56 + 0.5625 Reorder the terms: 0.5625 + 1.5b + b2 = -0.56 + 0.5625 Combine like terms: -0.56 + 0.5625 = 0.0025 0.5625 + 1.5b + b2 = 0.0025 Factor a perfect square on the left side: (b + 0.75)(b + 0.75) = 0.0025 Calculate the square root of the right side: 0.05 Break this problem into two subproblems by setting (b + 0.75) equal to 0.05 and -0.05.Subproblem 1
b + 0.75 = 0.05 Simplifying b + 0.75 = 0.05 Reorder the terms: 0.75 + b = 0.05 Solving 0.75 + b = 0.05 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + b = 0.05 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + b = 0.05 + -0.75 b = 0.05 + -0.75 Combine like terms: 0.05 + -0.75 = -0.7 b = -0.7 Simplifying b = -0.7Subproblem 2
b + 0.75 = -0.05 Simplifying b + 0.75 = -0.05 Reorder the terms: 0.75 + b = -0.05 Solving 0.75 + b = -0.05 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + b = -0.05 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + b = -0.05 + -0.75 b = -0.05 + -0.75 Combine like terms: -0.05 + -0.75 = -0.8 b = -0.8 Simplifying b = -0.8Solution
The solution to the problem is based on the solutions from the subproblems. b = {-0.7, -0.8}
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