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Simplifying (2b + -12)(b + 10) = 3b Reorder the terms: (-12 + 2b)(b + 10) = 3b Reorder the terms: (-12 + 2b)(10 + b) = 3b Multiply (-12 + 2b) * (10 + b) (-12(10 + b) + 2b * (10 + b)) = 3b ((10 * -12 + b * -12) + 2b * (10 + b)) = 3b ((-120 + -12b) + 2b * (10 + b)) = 3b (-120 + -12b + (10 * 2b + b * 2b)) = 3b (-120 + -12b + (20b + 2b2)) = 3b Combine like terms: -12b + 20b = 8b (-120 + 8b + 2b2) = 3b Solving -120 + 8b + 2b2 = 3b Solving for variable 'b'. Reorder the terms: -120 + 8b + -3b + 2b2 = 3b + -3b Combine like terms: 8b + -3b = 5b -120 + 5b + 2b2 = 3b + -3b Combine like terms: 3b + -3b = 0 -120 + 5b + 2b2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -60 + 2.5b + b2 = 0 Move the constant term to the right: Add '60' to each side of the equation. -60 + 2.5b + 60 + b2 = 0 + 60 Reorder the terms: -60 + 60 + 2.5b + b2 = 0 + 60 Combine like terms: -60 + 60 = 0 0 + 2.5b + b2 = 0 + 60 2.5b + b2 = 0 + 60 Combine like terms: 0 + 60 = 60 2.5b + b2 = 60 The b term is 2.5b. Take half its coefficient (1.25). Square it (1.5625) and add it to both sides. Add '1.5625' to each side of the equation. 2.5b + 1.5625 + b2 = 60 + 1.5625 Reorder the terms: 1.5625 + 2.5b + b2 = 60 + 1.5625 Combine like terms: 60 + 1.5625 = 61.5625 1.5625 + 2.5b + b2 = 61.5625 Factor a perfect square on the left side: (b + 1.25)(b + 1.25) = 61.5625 Calculate the square root of the right side: 7.846177413 Break this problem into two subproblems by setting (b + 1.25) equal to 7.846177413 and -7.846177413.Subproblem 1
b + 1.25 = 7.846177413 Simplifying b + 1.25 = 7.846177413 Reorder the terms: 1.25 + b = 7.846177413 Solving 1.25 + b = 7.846177413 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-1.25' to each side of the equation. 1.25 + -1.25 + b = 7.846177413 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + b = 7.846177413 + -1.25 b = 7.846177413 + -1.25 Combine like terms: 7.846177413 + -1.25 = 6.596177413 b = 6.596177413 Simplifying b = 6.596177413Subproblem 2
b + 1.25 = -7.846177413 Simplifying b + 1.25 = -7.846177413 Reorder the terms: 1.25 + b = -7.846177413 Solving 1.25 + b = -7.846177413 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-1.25' to each side of the equation. 1.25 + -1.25 + b = -7.846177413 + -1.25 Combine like terms: 1.25 + -1.25 = 0.00 0.00 + b = -7.846177413 + -1.25 b = -7.846177413 + -1.25 Combine like terms: -7.846177413 + -1.25 = -9.096177413 b = -9.096177413 Simplifying b = -9.096177413Solution
The solution to the problem is based on the solutions from the subproblems. b = {6.596177413, -9.096177413}
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