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Simplifying (2b2) + 3b = 1 Reorder the terms: 3b + (2b2) = 1 Solving 3b + (2b2) = 1 Solving for variable 'b'. Reorder the terms: -1 + 3b + (2b2) = 1 + -1 Combine like terms: 1 + -1 = 0 -1 + 3b + (2b2) = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -0.5 + 1.5b + b2 = 0 Move the constant term to the right: Add '0.5' to each side of the equation. -0.5 + 1.5b + 0.5 + b2 = 0 + 0.5 Reorder the terms: -0.5 + 0.5 + 1.5b + b2 = 0 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + 1.5b + b2 = 0 + 0.5 1.5b + b2 = 0 + 0.5 Combine like terms: 0 + 0.5 = 0.5 1.5b + b2 = 0.5 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 = 0.5 + 0.5625 Reorder the terms: 0.5625 + 1.5b + b2 = 0.5 + 0.5625 Combine like terms: 0.5 + 0.5625 = 1.0625 0.5625 + 1.5b + b2 = 1.0625 Factor a perfect square on the left side: ((b) + 0.75)((b) + 0.75) = 1.0625 Calculate the square root of the right side: 1.030776406 Break this problem into two subproblems by setting ((b) + 0.75) equal to 1.030776406 and -1.030776406.Subproblem 1
(b) + 0.75 = 1.030776406 Simplifying (b) + 0.75 = 1.030776406 b + 0.75 = 1.030776406 Reorder the terms: 0.75 + b = 1.030776406 Solving 0.75 + b = 1.030776406 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 = 1.030776406 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + b = 1.030776406 + -0.75 b = 1.030776406 + -0.75 Combine like terms: 1.030776406 + -0.75 = 0.280776406 b = 0.280776406 Simplifying b = 0.280776406Subproblem 2
(b) + 0.75 = -1.030776406 Simplifying (b) + 0.75 = -1.030776406 b + 0.75 = -1.030776406 Reorder the terms: 0.75 + b = -1.030776406 Solving 0.75 + b = -1.030776406 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 = -1.030776406 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + b = -1.030776406 + -0.75 b = -1.030776406 + -0.75 Combine like terms: -1.030776406 + -0.75 = -1.780776406 b = -1.780776406 Simplifying b = -1.780776406Solution
The solution to the problem is based on the solutions from the subproblems. b = {0.280776406, -1.780776406}
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