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Simplifying 21b2 + -48b + -46 = 0 Reorder the terms: -46 + -48b + 21b2 = 0 Solving -46 + -48b + 21b2 = 0 Solving for variable 'b'. Begin completing the square. Divide all terms by 21 the coefficient of the squared term: Divide each side by '21'. -2.19047619 + -2.285714286b + b2 = 0 Move the constant term to the right: Add '2.19047619' to each side of the equation. -2.19047619 + -2.285714286b + 2.19047619 + b2 = 0 + 2.19047619 Reorder the terms: -2.19047619 + 2.19047619 + -2.285714286b + b2 = 0 + 2.19047619 Combine like terms: -2.19047619 + 2.19047619 = 0.00000000 0.00000000 + -2.285714286b + b2 = 0 + 2.19047619 -2.285714286b + b2 = 0 + 2.19047619 Combine like terms: 0 + 2.19047619 = 2.19047619 -2.285714286b + b2 = 2.19047619 The b term is -2.285714286b. Take half its coefficient (-1.142857143). Square it (1.306122449) and add it to both sides. Add '1.306122449' to each side of the equation. -2.285714286b + 1.306122449 + b2 = 2.19047619 + 1.306122449 Reorder the terms: 1.306122449 + -2.285714286b + b2 = 2.19047619 + 1.306122449 Combine like terms: 2.19047619 + 1.306122449 = 3.496598639 1.306122449 + -2.285714286b + b2 = 3.496598639 Factor a perfect square on the left side: (b + -1.142857143)(b + -1.142857143) = 3.496598639 Calculate the square root of the right side: 1.86991942 Break this problem into two subproblems by setting (b + -1.142857143) equal to 1.86991942 and -1.86991942.Subproblem 1
b + -1.142857143 = 1.86991942 Simplifying b + -1.142857143 = 1.86991942 Reorder the terms: -1.142857143 + b = 1.86991942 Solving -1.142857143 + b = 1.86991942 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '1.142857143' to each side of the equation. -1.142857143 + 1.142857143 + b = 1.86991942 + 1.142857143 Combine like terms: -1.142857143 + 1.142857143 = 0.000000000 0.000000000 + b = 1.86991942 + 1.142857143 b = 1.86991942 + 1.142857143 Combine like terms: 1.86991942 + 1.142857143 = 3.012776563 b = 3.012776563 Simplifying b = 3.012776563Subproblem 2
b + -1.142857143 = -1.86991942 Simplifying b + -1.142857143 = -1.86991942 Reorder the terms: -1.142857143 + b = -1.86991942 Solving -1.142857143 + b = -1.86991942 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '1.142857143' to each side of the equation. -1.142857143 + 1.142857143 + b = -1.86991942 + 1.142857143 Combine like terms: -1.142857143 + 1.142857143 = 0.000000000 0.000000000 + b = -1.86991942 + 1.142857143 b = -1.86991942 + 1.142857143 Combine like terms: -1.86991942 + 1.142857143 = -0.727062277 b = -0.727062277 Simplifying b = -0.727062277Solution
The solution to the problem is based on the solutions from the subproblems. b = {3.012776563, -0.727062277}
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