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Simplifying -2b2 + -11b = -23 Reorder the terms: -11b + -2b2 = -23 Solving -11b + -2b2 = -23 Solving for variable 'b'. Reorder the terms: 23 + -11b + -2b2 = -23 + 23 Combine like terms: -23 + 23 = 0 23 + -11b + -2b2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -11.5 + 5.5b + b2 = 0 Move the constant term to the right: Add '11.5' to each side of the equation. -11.5 + 5.5b + 11.5 + b2 = 0 + 11.5 Reorder the terms: -11.5 + 11.5 + 5.5b + b2 = 0 + 11.5 Combine like terms: -11.5 + 11.5 = 0.0 0.0 + 5.5b + b2 = 0 + 11.5 5.5b + b2 = 0 + 11.5 Combine like terms: 0 + 11.5 = 11.5 5.5b + b2 = 11.5 The b term is 5.5b. Take half its coefficient (2.75). Square it (7.5625) and add it to both sides. Add '7.5625' to each side of the equation. 5.5b + 7.5625 + b2 = 11.5 + 7.5625 Reorder the terms: 7.5625 + 5.5b + b2 = 11.5 + 7.5625 Combine like terms: 11.5 + 7.5625 = 19.0625 7.5625 + 5.5b + b2 = 19.0625 Factor a perfect square on the left side: (b + 2.75)(b + 2.75) = 19.0625 Calculate the square root of the right side: 4.366062299 Break this problem into two subproblems by setting (b + 2.75) equal to 4.366062299 and -4.366062299.Subproblem 1
b + 2.75 = 4.366062299 Simplifying b + 2.75 = 4.366062299 Reorder the terms: 2.75 + b = 4.366062299 Solving 2.75 + b = 4.366062299 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-2.75' to each side of the equation. 2.75 + -2.75 + b = 4.366062299 + -2.75 Combine like terms: 2.75 + -2.75 = 0.00 0.00 + b = 4.366062299 + -2.75 b = 4.366062299 + -2.75 Combine like terms: 4.366062299 + -2.75 = 1.616062299 b = 1.616062299 Simplifying b = 1.616062299Subproblem 2
b + 2.75 = -4.366062299 Simplifying b + 2.75 = -4.366062299 Reorder the terms: 2.75 + b = -4.366062299 Solving 2.75 + b = -4.366062299 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-2.75' to each side of the equation. 2.75 + -2.75 + b = -4.366062299 + -2.75 Combine like terms: 2.75 + -2.75 = 0.00 0.00 + b = -4.366062299 + -2.75 b = -4.366062299 + -2.75 Combine like terms: -4.366062299 + -2.75 = -7.116062299 b = -7.116062299 Simplifying b = -7.116062299Solution
The solution to the problem is based on the solutions from the subproblems. b = {1.616062299, -7.116062299}
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