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Simplifying 2b2 + -5b = 23 Reorder the terms: -5b + 2b2 = 23 Solving -5b + 2b2 = 23 Solving for variable 'b'. Reorder the terms: -23 + -5b + 2b2 = 23 + -23 Combine like terms: 23 + -23 = 0 -23 + -5b + 2b2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -11.5 + -2.5b + b2 = 0 Move the constant term to the right: Add '11.5' to each side of the equation. -11.5 + -2.5b + 11.5 + b2 = 0 + 11.5 Reorder the terms: -11.5 + 11.5 + -2.5b + b2 = 0 + 11.5 Combine like terms: -11.5 + 11.5 = 0.0 0.0 + -2.5b + b2 = 0 + 11.5 -2.5b + b2 = 0 + 11.5 Combine like terms: 0 + 11.5 = 11.5 -2.5b + b2 = 11.5 The b term is -2.5b. Take half its coefficient (-1.25). Square it (1.5625) and add it to both sides. Add '1.5625' to each side of the equation. -2.5b + 1.5625 + b2 = 11.5 + 1.5625 Reorder the terms: 1.5625 + -2.5b + b2 = 11.5 + 1.5625 Combine like terms: 11.5 + 1.5625 = 13.0625 1.5625 + -2.5b + b2 = 13.0625 Factor a perfect square on the left side: (b + -1.25)(b + -1.25) = 13.0625 Calculate the square root of the right side: 3.614208074 Break this problem into two subproblems by setting (b + -1.25) equal to 3.614208074 and -3.614208074.Subproblem 1
b + -1.25 = 3.614208074 Simplifying b + -1.25 = 3.614208074 Reorder the terms: -1.25 + b = 3.614208074 Solving -1.25 + b = 3.614208074 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + b = 3.614208074 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + b = 3.614208074 + 1.25 b = 3.614208074 + 1.25 Combine like terms: 3.614208074 + 1.25 = 4.864208074 b = 4.864208074 Simplifying b = 4.864208074Subproblem 2
b + -1.25 = -3.614208074 Simplifying b + -1.25 = -3.614208074 Reorder the terms: -1.25 + b = -3.614208074 Solving -1.25 + b = -3.614208074 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + b = -3.614208074 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + b = -3.614208074 + 1.25 b = -3.614208074 + 1.25 Combine like terms: -3.614208074 + 1.25 = -2.364208074 b = -2.364208074 Simplifying b = -2.364208074Solution
The solution to the problem is based on the solutions from the subproblems. b = {4.864208074, -2.364208074}
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