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Simplifying 6b2 = 6b + 23 Reorder the terms: 6b2 = 23 + 6b Solving 6b2 = 23 + 6b Solving for variable 'b'. Reorder the terms: -23 + -6b + 6b2 = 23 + 6b + -23 + -6b Reorder the terms: -23 + -6b + 6b2 = 23 + -23 + 6b + -6b Combine like terms: 23 + -23 = 0 -23 + -6b + 6b2 = 0 + 6b + -6b -23 + -6b + 6b2 = 6b + -6b Combine like terms: 6b + -6b = 0 -23 + -6b + 6b2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -3.833333333 + -1b + b2 = 0 Move the constant term to the right: Add '3.833333333' to each side of the equation. -3.833333333 + -1b + 3.833333333 + b2 = 0 + 3.833333333 Reorder the terms: -3.833333333 + 3.833333333 + -1b + b2 = 0 + 3.833333333 Combine like terms: -3.833333333 + 3.833333333 = 0.000000000 0.000000000 + -1b + b2 = 0 + 3.833333333 -1b + b2 = 0 + 3.833333333 Combine like terms: 0 + 3.833333333 = 3.833333333 -1b + b2 = 3.833333333 The b term is -1b. Take half its coefficient (-0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. -1b + 0.25 + b2 = 3.833333333 + 0.25 Reorder the terms: 0.25 + -1b + b2 = 3.833333333 + 0.25 Combine like terms: 3.833333333 + 0.25 = 4.083333333 0.25 + -1b + b2 = 4.083333333 Factor a perfect square on the left side: (b + -0.5)(b + -0.5) = 4.083333333 Calculate the square root of the right side: 2.020725942 Break this problem into two subproblems by setting (b + -0.5) equal to 2.020725942 and -2.020725942.Subproblem 1
b + -0.5 = 2.020725942 Simplifying b + -0.5 = 2.020725942 Reorder the terms: -0.5 + b = 2.020725942 Solving -0.5 + b = 2.020725942 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + b = 2.020725942 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + b = 2.020725942 + 0.5 b = 2.020725942 + 0.5 Combine like terms: 2.020725942 + 0.5 = 2.520725942 b = 2.520725942 Simplifying b = 2.520725942Subproblem 2
b + -0.5 = -2.020725942 Simplifying b + -0.5 = -2.020725942 Reorder the terms: -0.5 + b = -2.020725942 Solving -0.5 + b = -2.020725942 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '0.5' to each side of the equation. -0.5 + 0.5 + b = -2.020725942 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + b = -2.020725942 + 0.5 b = -2.020725942 + 0.5 Combine like terms: -2.020725942 + 0.5 = -1.520725942 b = -1.520725942 Simplifying b = -1.520725942Solution
The solution to the problem is based on the solutions from the subproblems. b = {2.520725942, -1.520725942}
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