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Simplifying 8b2 + -8b = 23 Reorder the terms: -8b + 8b2 = 23 Solving -8b + 8b2 = 23 Solving for variable 'b'. Reorder the terms: -23 + -8b + 8b2 = 23 + -23 Combine like terms: 23 + -23 = 0 -23 + -8b + 8b2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -2.875 + -1b + b2 = 0 Move the constant term to the right: Add '2.875' to each side of the equation. -2.875 + -1b + 2.875 + b2 = 0 + 2.875 Reorder the terms: -2.875 + 2.875 + -1b + b2 = 0 + 2.875 Combine like terms: -2.875 + 2.875 = 0.000 0.000 + -1b + b2 = 0 + 2.875 -1b + b2 = 0 + 2.875 Combine like terms: 0 + 2.875 = 2.875 -1b + b2 = 2.875 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 = 2.875 + 0.25 Reorder the terms: 0.25 + -1b + b2 = 2.875 + 0.25 Combine like terms: 2.875 + 0.25 = 3.125 0.25 + -1b + b2 = 3.125 Factor a perfect square on the left side: (b + -0.5)(b + -0.5) = 3.125 Calculate the square root of the right side: 1.767766953 Break this problem into two subproblems by setting (b + -0.5) equal to 1.767766953 and -1.767766953.Subproblem 1
b + -0.5 = 1.767766953 Simplifying b + -0.5 = 1.767766953 Reorder the terms: -0.5 + b = 1.767766953 Solving -0.5 + b = 1.767766953 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 = 1.767766953 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + b = 1.767766953 + 0.5 b = 1.767766953 + 0.5 Combine like terms: 1.767766953 + 0.5 = 2.267766953 b = 2.267766953 Simplifying b = 2.267766953Subproblem 2
b + -0.5 = -1.767766953 Simplifying b + -0.5 = -1.767766953 Reorder the terms: -0.5 + b = -1.767766953 Solving -0.5 + b = -1.767766953 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 = -1.767766953 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + b = -1.767766953 + 0.5 b = -1.767766953 + 0.5 Combine like terms: -1.767766953 + 0.5 = -1.267766953 b = -1.267766953 Simplifying b = -1.267766953Solution
The solution to the problem is based on the solutions from the subproblems. b = {2.267766953, -1.267766953}
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