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Simplifying 0.5b(b + 2) = 80 Reorder the terms: 0.5b(2 + b) = 80 (2 * 0.5b + b * 0.5b) = 80 (1b + 0.5b2) = 80 Solving 1b + 0.5b2 = 80 Solving for variable 'b'. Reorder the terms: -80 + 1b + 0.5b2 = 80 + -80 Combine like terms: 80 + -80 = 0 -80 + 1b + 0.5b2 = 0 Begin completing the square. Divide all terms by 0.5 the coefficient of the squared term: Divide each side by '0.5'. -160 + 2b + b2 = 0 Move the constant term to the right: Add '160' to each side of the equation. -160 + 2b + 160 + b2 = 0 + 160 Reorder the terms: -160 + 160 + 2b + b2 = 0 + 160 Combine like terms: -160 + 160 = 0 0 + 2b + b2 = 0 + 160 2b + b2 = 0 + 160 Combine like terms: 0 + 160 = 160 2b + b2 = 160 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 = 160 + 1 Reorder the terms: 1 + 2b + b2 = 160 + 1 Combine like terms: 160 + 1 = 161 1 + 2b + b2 = 161 Factor a perfect square on the left side: (b + 1)(b + 1) = 161 Calculate the square root of the right side: 12.68857754 Break this problem into two subproblems by setting (b + 1) equal to 12.68857754 and -12.68857754.Subproblem 1
b + 1 = 12.68857754 Simplifying b + 1 = 12.68857754 Reorder the terms: 1 + b = 12.68857754 Solving 1 + b = 12.68857754 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 = 12.68857754 + -1 Combine like terms: 1 + -1 = 0 0 + b = 12.68857754 + -1 b = 12.68857754 + -1 Combine like terms: 12.68857754 + -1 = 11.68857754 b = 11.68857754 Simplifying b = 11.68857754Subproblem 2
b + 1 = -12.68857754 Simplifying b + 1 = -12.68857754 Reorder the terms: 1 + b = -12.68857754 Solving 1 + b = -12.68857754 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 = -12.68857754 + -1 Combine like terms: 1 + -1 = 0 0 + b = -12.68857754 + -1 b = -12.68857754 + -1 Combine like terms: -12.68857754 + -1 = -13.68857754 b = -13.68857754 Simplifying b = -13.68857754Solution
The solution to the problem is based on the solutions from the subproblems. b = {11.68857754, -13.68857754}
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