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Simplifying -1b2 + -2b + 1 = 0 Reorder the terms: 1 + -2b + -1b2 = 0 Solving 1 + -2b + -1b2 = 0 Solving for variable 'b'. Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -1 + 2b + b2 = 0 Move the constant term to the right: Add '1' to each side of the equation. -1 + 2b + 1 + b2 = 0 + 1 Reorder the terms: -1 + 1 + 2b + b2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + 2b + b2 = 0 + 1 2b + b2 = 0 + 1 Combine like terms: 0 + 1 = 1 2b + b2 = 1 The b term is 2b. Take half its coefficient (1). Square it (1) and add it to both sides. Add '1' to each side of the equation. 2b + 1 + b2 = 1 + 1 Reorder the terms: 1 + 2b + b2 = 1 + 1 Combine like terms: 1 + 1 = 2 1 + 2b + b2 = 2 Factor a perfect square on the left side: (b + 1)(b + 1) = 2 Calculate the square root of the right side: 1.414213562 Break this problem into two subproblems by setting (b + 1) equal to 1.414213562 and -1.414213562.Subproblem 1
b + 1 = 1.414213562 Simplifying b + 1 = 1.414213562 Reorder the terms: 1 + b = 1.414213562 Solving 1 + b = 1.414213562 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + b = 1.414213562 + -1 Combine like terms: 1 + -1 = 0 0 + b = 1.414213562 + -1 b = 1.414213562 + -1 Combine like terms: 1.414213562 + -1 = 0.414213562 b = 0.414213562 Simplifying b = 0.414213562Subproblem 2
b + 1 = -1.414213562 Simplifying b + 1 = -1.414213562 Reorder the terms: 1 + b = -1.414213562 Solving 1 + b = -1.414213562 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + b = -1.414213562 + -1 Combine like terms: 1 + -1 = 0 0 + b = -1.414213562 + -1 b = -1.414213562 + -1 Combine like terms: -1.414213562 + -1 = -2.414213562 b = -2.414213562 Simplifying b = -2.414213562Solution
The solution to the problem is based on the solutions from the subproblems. b = {0.414213562, -2.414213562}
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