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Simplifying 4b2 + 40b = -82 Reorder the terms: 40b + 4b2 = -82 Solving 40b + 4b2 = -82 Solving for variable 'b'. Reorder the terms: 82 + 40b + 4b2 = -82 + 82 Combine like terms: -82 + 82 = 0 82 + 40b + 4b2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(41 + 20b + 2b2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(41 + 20b + 2b2)' equal to zero and attempt to solve: Simplifying 41 + 20b + 2b2 = 0 Solving 41 + 20b + 2b2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. 20.5 + 10b + b2 = 0 Move the constant term to the right: Add '-20.5' to each side of the equation. 20.5 + 10b + -20.5 + b2 = 0 + -20.5 Reorder the terms: 20.5 + -20.5 + 10b + b2 = 0 + -20.5 Combine like terms: 20.5 + -20.5 = 0.0 0.0 + 10b + b2 = 0 + -20.5 10b + b2 = 0 + -20.5 Combine like terms: 0 + -20.5 = -20.5 10b + b2 = -20.5 The b term is 10b. Take half its coefficient (5). Square it (25) and add it to both sides. Add '25' to each side of the equation. 10b + 25 + b2 = -20.5 + 25 Reorder the terms: 25 + 10b + b2 = -20.5 + 25 Combine like terms: -20.5 + 25 = 4.5 25 + 10b + b2 = 4.5 Factor a perfect square on the left side: (b + 5)(b + 5) = 4.5 Calculate the square root of the right side: 2.121320344 Break this problem into two subproblems by setting (b + 5) equal to 2.121320344 and -2.121320344.Subproblem 1
b + 5 = 2.121320344 Simplifying b + 5 = 2.121320344 Reorder the terms: 5 + b = 2.121320344 Solving 5 + b = 2.121320344 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-5' to each side of the equation. 5 + -5 + b = 2.121320344 + -5 Combine like terms: 5 + -5 = 0 0 + b = 2.121320344 + -5 b = 2.121320344 + -5 Combine like terms: 2.121320344 + -5 = -2.878679656 b = -2.878679656 Simplifying b = -2.878679656Subproblem 2
b + 5 = -2.121320344 Simplifying b + 5 = -2.121320344 Reorder the terms: 5 + b = -2.121320344 Solving 5 + b = -2.121320344 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-5' to each side of the equation. 5 + -5 + b = -2.121320344 + -5 Combine like terms: 5 + -5 = 0 0 + b = -2.121320344 + -5 b = -2.121320344 + -5 Combine like terms: -2.121320344 + -5 = -7.121320344 b = -7.121320344 Simplifying b = -7.121320344Solution
The solution to the problem is based on the solutions from the subproblems. b = {-2.878679656, -7.121320344}Solution
b = {-2.878679656, -7.121320344}
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