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Simplifying 4b2 + 23b = -23 Reorder the terms: 23b + 4b2 = -23 Solving 23b + 4b2 = -23 Solving for variable 'b'. Reorder the terms: 23 + 23b + 4b2 = -23 + 23 Combine like terms: -23 + 23 = 0 23 + 23b + 4b2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. 5.75 + 5.75b + b2 = 0 Move the constant term to the right: Add '-5.75' to each side of the equation. 5.75 + 5.75b + -5.75 + b2 = 0 + -5.75 Reorder the terms: 5.75 + -5.75 + 5.75b + b2 = 0 + -5.75 Combine like terms: 5.75 + -5.75 = 0.00 0.00 + 5.75b + b2 = 0 + -5.75 5.75b + b2 = 0 + -5.75 Combine like terms: 0 + -5.75 = -5.75 5.75b + b2 = -5.75 The b term is 5.75b. Take half its coefficient (2.875). Square it (8.265625) and add it to both sides. Add '8.265625' to each side of the equation. 5.75b + 8.265625 + b2 = -5.75 + 8.265625 Reorder the terms: 8.265625 + 5.75b + b2 = -5.75 + 8.265625 Combine like terms: -5.75 + 8.265625 = 2.515625 8.265625 + 5.75b + b2 = 2.515625 Factor a perfect square on the left side: (b + 2.875)(b + 2.875) = 2.515625 Calculate the square root of the right side: 1.586072193 Break this problem into two subproblems by setting (b + 2.875) equal to 1.586072193 and -1.586072193.Subproblem 1
b + 2.875 = 1.586072193 Simplifying b + 2.875 = 1.586072193 Reorder the terms: 2.875 + b = 1.586072193 Solving 2.875 + b = 1.586072193 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-2.875' to each side of the equation. 2.875 + -2.875 + b = 1.586072193 + -2.875 Combine like terms: 2.875 + -2.875 = 0.000 0.000 + b = 1.586072193 + -2.875 b = 1.586072193 + -2.875 Combine like terms: 1.586072193 + -2.875 = -1.288927807 b = -1.288927807 Simplifying b = -1.288927807Subproblem 2
b + 2.875 = -1.586072193 Simplifying b + 2.875 = -1.586072193 Reorder the terms: 2.875 + b = -1.586072193 Solving 2.875 + b = -1.586072193 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-2.875' to each side of the equation. 2.875 + -2.875 + b = -1.586072193 + -2.875 Combine like terms: 2.875 + -2.875 = 0.000 0.000 + b = -1.586072193 + -2.875 b = -1.586072193 + -2.875 Combine like terms: -1.586072193 + -2.875 = -4.461072193 b = -4.461072193 Simplifying b = -4.461072193Solution
The solution to the problem is based on the solutions from the subproblems. b = {-1.288927807, -4.461072193}
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