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Simplifying 0 = b2 + 1.382b + -1.654 Reorder the terms: 0 = -1.654 + 1.382b + b2 Solving 0 = -1.654 + 1.382b + b2 Solving for variable 'b'. Combine like terms: 0 + 1.654 = 1.654 1.654 + -1.382b + -1b2 = -1.654 + 1.382b + b2 + 1.654 + -1.382b + -1b2 Reorder the terms: 1.654 + -1.382b + -1b2 = -1.654 + 1.654 + 1.382b + -1.382b + b2 + -1b2 Combine like terms: -1.654 + 1.654 = 0.000 1.654 + -1.382b + -1b2 = 0.000 + 1.382b + -1.382b + b2 + -1b2 1.654 + -1.382b + -1b2 = 1.382b + -1.382b + b2 + -1b2 Combine like terms: 1.382b + -1.382b = 0.000 1.654 + -1.382b + -1b2 = 0.000 + b2 + -1b2 1.654 + -1.382b + -1b2 = b2 + -1b2 Combine like terms: b2 + -1b2 = 0 1.654 + -1.382b + -1b2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -1.654 + 1.382b + b2 = 0 Move the constant term to the right: Add '1.654' to each side of the equation. -1.654 + 1.382b + 1.654 + b2 = 0 + 1.654 Reorder the terms: -1.654 + 1.654 + 1.382b + b2 = 0 + 1.654 Combine like terms: -1.654 + 1.654 = 0.000 0.000 + 1.382b + b2 = 0 + 1.654 1.382b + b2 = 0 + 1.654 Combine like terms: 0 + 1.654 = 1.654 1.382b + b2 = 1.654 The b term is 1.382b. Take half its coefficient (0.691). Square it (0.477481) and add it to both sides. Add '0.477481' to each side of the equation. 1.382b + 0.477481 + b2 = 1.654 + 0.477481 Reorder the terms: 0.477481 + 1.382b + b2 = 1.654 + 0.477481 Combine like terms: 1.654 + 0.477481 = 2.131481 0.477481 + 1.382b + b2 = 2.131481 Factor a perfect square on the left side: (b + 0.691)(b + 0.691) = 2.131481 Calculate the square root of the right side: 1.459959246 Break this problem into two subproblems by setting (b + 0.691) equal to 1.459959246 and -1.459959246.Subproblem 1
b + 0.691 = 1.459959246 Simplifying b + 0.691 = 1.459959246 Reorder the terms: 0.691 + b = 1.459959246 Solving 0.691 + b = 1.459959246 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-0.691' to each side of the equation. 0.691 + -0.691 + b = 1.459959246 + -0.691 Combine like terms: 0.691 + -0.691 = 0.000 0.000 + b = 1.459959246 + -0.691 b = 1.459959246 + -0.691 Combine like terms: 1.459959246 + -0.691 = 0.768959246 b = 0.768959246 Simplifying b = 0.768959246Subproblem 2
b + 0.691 = -1.459959246 Simplifying b + 0.691 = -1.459959246 Reorder the terms: 0.691 + b = -1.459959246 Solving 0.691 + b = -1.459959246 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-0.691' to each side of the equation. 0.691 + -0.691 + b = -1.459959246 + -0.691 Combine like terms: 0.691 + -0.691 = 0.000 0.000 + b = -1.459959246 + -0.691 b = -1.459959246 + -0.691 Combine like terms: -1.459959246 + -0.691 = -2.150959246 b = -2.150959246 Simplifying b = -2.150959246Solution
The solution to the problem is based on the solutions from the subproblems. b = {0.768959246, -2.150959246}
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