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Simplifying 31500 = b2 + 30b Reorder the terms: 31500 = 30b + b2 Solving 31500 = 30b + b2 Solving for variable 'b'. Reorder the terms: 31500 + -30b + -1b2 = 30b + -30b + b2 + -1b2 Combine like terms: 30b + -30b = 0 31500 + -30b + -1b2 = 0 + b2 + -1b2 31500 + -30b + -1b2 = b2 + -1b2 Combine like terms: b2 + -1b2 = 0 31500 + -30b + -1b2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -31500 + 30b + b2 = 0 Move the constant term to the right: Add '31500' to each side of the equation. -31500 + 30b + 31500 + b2 = 0 + 31500 Reorder the terms: -31500 + 31500 + 30b + b2 = 0 + 31500 Combine like terms: -31500 + 31500 = 0 0 + 30b + b2 = 0 + 31500 30b + b2 = 0 + 31500 Combine like terms: 0 + 31500 = 31500 30b + b2 = 31500 The b term is 30b. Take half its coefficient (15). Square it (225) and add it to both sides. Add '225' to each side of the equation. 30b + 225 + b2 = 31500 + 225 Reorder the terms: 225 + 30b + b2 = 31500 + 225 Combine like terms: 31500 + 225 = 31725 225 + 30b + b2 = 31725 Factor a perfect square on the left side: (b + 15)(b + 15) = 31725 Calculate the square root of the right side: 178.115131306 Break this problem into two subproblems by setting (b + 15) equal to 178.115131306 and -178.115131306.Subproblem 1
b + 15 = 178.115131306 Simplifying b + 15 = 178.115131306 Reorder the terms: 15 + b = 178.115131306 Solving 15 + b = 178.115131306 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-15' to each side of the equation. 15 + -15 + b = 178.115131306 + -15 Combine like terms: 15 + -15 = 0 0 + b = 178.115131306 + -15 b = 178.115131306 + -15 Combine like terms: 178.115131306 + -15 = 163.115131306 b = 163.115131306 Simplifying b = 163.115131306Subproblem 2
b + 15 = -178.115131306 Simplifying b + 15 = -178.115131306 Reorder the terms: 15 + b = -178.115131306 Solving 15 + b = -178.115131306 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-15' to each side of the equation. 15 + -15 + b = -178.115131306 + -15 Combine like terms: 15 + -15 = 0 0 + b = -178.115131306 + -15 b = -178.115131306 + -15 Combine like terms: -178.115131306 + -15 = -193.115131306 b = -193.115131306 Simplifying b = -193.115131306Solution
The solution to the problem is based on the solutions from the subproblems. b = {163.115131306, -193.115131306}
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