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Simplifying (0.6b + -0.1)(0.11b + 1) = 0 Reorder the terms: (-0.1 + 0.6b)(0.11b + 1) = 0 Reorder the terms: (-0.1 + 0.6b)(1 + 0.11b) = 0 Multiply (-0.1 + 0.6b) * (1 + 0.11b) (-0.1(1 + 0.11b) + 0.6b * (1 + 0.11b)) = 0 ((1 * -0.1 + 0.11b * -0.1) + 0.6b * (1 + 0.11b)) = 0 ((-0.1 + -0.011b) + 0.6b * (1 + 0.11b)) = 0 (-0.1 + -0.011b + (1 * 0.6b + 0.11b * 0.6b)) = 0 (-0.1 + -0.011b + (0.6b + 0.066b2)) = 0 Combine like terms: -0.011b + 0.6b = 0.589b (-0.1 + 0.589b + 0.066b2) = 0 Solving -0.1 + 0.589b + 0.066b2 = 0 Solving for variable 'b'. Begin completing the square. Divide all terms by 0.066 the coefficient of the squared term: Divide each side by '0.066'. -1.515151515 + 8.924242424b + b2 = 0 Move the constant term to the right: Add '1.515151515' to each side of the equation. -1.515151515 + 8.924242424b + 1.515151515 + b2 = 0 + 1.515151515 Reorder the terms: -1.515151515 + 1.515151515 + 8.924242424b + b2 = 0 + 1.515151515 Combine like terms: -1.515151515 + 1.515151515 = 0.000000000 0.000000000 + 8.924242424b + b2 = 0 + 1.515151515 8.924242424b + b2 = 0 + 1.515151515 Combine like terms: 0 + 1.515151515 = 1.515151515 8.924242424b + b2 = 1.515151515 The b term is 8.924242424b. Take half its coefficient (4.462121212). Square it (19.91052571) and add it to both sides. Add '19.91052571' to each side of the equation. 8.924242424b + 19.91052571 + b2 = 1.515151515 + 19.91052571 Reorder the terms: 19.91052571 + 8.924242424b + b2 = 1.515151515 + 19.91052571 Combine like terms: 1.515151515 + 19.91052571 = 21.425677225 19.91052571 + 8.924242424b + b2 = 21.425677225 Factor a perfect square on the left side: (b + 4.462121212)(b + 4.462121212) = 21.425677225 Calculate the square root of the right side: 4.628787879 Break this problem into two subproblems by setting (b + 4.462121212) equal to 4.628787879 and -4.628787879.Subproblem 1
b + 4.462121212 = 4.628787879 Simplifying b + 4.462121212 = 4.628787879 Reorder the terms: 4.462121212 + b = 4.628787879 Solving 4.462121212 + b = 4.628787879 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-4.462121212' to each side of the equation. 4.462121212 + -4.462121212 + b = 4.628787879 + -4.462121212 Combine like terms: 4.462121212 + -4.462121212 = 0.000000000 0.000000000 + b = 4.628787879 + -4.462121212 b = 4.628787879 + -4.462121212 Combine like terms: 4.628787879 + -4.462121212 = 0.166666667 b = 0.166666667 Simplifying b = 0.166666667Subproblem 2
b + 4.462121212 = -4.628787879 Simplifying b + 4.462121212 = -4.628787879 Reorder the terms: 4.462121212 + b = -4.628787879 Solving 4.462121212 + b = -4.628787879 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-4.462121212' to each side of the equation. 4.462121212 + -4.462121212 + b = -4.628787879 + -4.462121212 Combine like terms: 4.462121212 + -4.462121212 = 0.000000000 0.000000000 + b = -4.628787879 + -4.462121212 b = -4.628787879 + -4.462121212 Combine like terms: -4.628787879 + -4.462121212 = -9.090909091 b = -9.090909091 Simplifying b = -9.090909091Solution
The solution to the problem is based on the solutions from the subproblems. b = {0.166666667, -9.090909091}
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