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Simplifying 10b2 + -41b + 5 = 0 Reorder the terms: 5 + -41b + 10b2 = 0 Solving 5 + -41b + 10b2 = 0 Solving for variable 'b'. Begin completing the square. Divide all terms by 10 the coefficient of the squared term: Divide each side by '10'. 0.5 + -4.1b + b2 = 0 Move the constant term to the right: Add '-0.5' to each side of the equation. 0.5 + -4.1b + -0.5 + b2 = 0 + -0.5 Reorder the terms: 0.5 + -0.5 + -4.1b + b2 = 0 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + -4.1b + b2 = 0 + -0.5 -4.1b + b2 = 0 + -0.5 Combine like terms: 0 + -0.5 = -0.5 -4.1b + b2 = -0.5 The b term is -4.1b. Take half its coefficient (-2.05). Square it (4.2025) and add it to both sides. Add '4.2025' to each side of the equation. -4.1b + 4.2025 + b2 = -0.5 + 4.2025 Reorder the terms: 4.2025 + -4.1b + b2 = -0.5 + 4.2025 Combine like terms: -0.5 + 4.2025 = 3.7025 4.2025 + -4.1b + b2 = 3.7025 Factor a perfect square on the left side: (b + -2.05)(b + -2.05) = 3.7025 Calculate the square root of the right side: 1.92418814 Break this problem into two subproblems by setting (b + -2.05) equal to 1.92418814 and -1.92418814.Subproblem 1
b + -2.05 = 1.92418814 Simplifying b + -2.05 = 1.92418814 Reorder the terms: -2.05 + b = 1.92418814 Solving -2.05 + b = 1.92418814 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '2.05' to each side of the equation. -2.05 + 2.05 + b = 1.92418814 + 2.05 Combine like terms: -2.05 + 2.05 = 0.00 0.00 + b = 1.92418814 + 2.05 b = 1.92418814 + 2.05 Combine like terms: 1.92418814 + 2.05 = 3.97418814 b = 3.97418814 Simplifying b = 3.97418814Subproblem 2
b + -2.05 = -1.92418814 Simplifying b + -2.05 = -1.92418814 Reorder the terms: -2.05 + b = -1.92418814 Solving -2.05 + b = -1.92418814 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '2.05' to each side of the equation. -2.05 + 2.05 + b = -1.92418814 + 2.05 Combine like terms: -2.05 + 2.05 = 0.00 0.00 + b = -1.92418814 + 2.05 b = -1.92418814 + 2.05 Combine like terms: -1.92418814 + 2.05 = 0.12581186 b = 0.12581186 Simplifying b = 0.12581186Solution
The solution to the problem is based on the solutions from the subproblems. b = {3.97418814, 0.12581186}
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