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Simplifying 2b2 + 28b + 196 = 1225 Reorder the terms: 196 + 28b + 2b2 = 1225 Solving 196 + 28b + 2b2 = 1225 Solving for variable 'b'. Reorder the terms: 196 + -1225 + 28b + 2b2 = 1225 + -1225 Combine like terms: 196 + -1225 = -1029 -1029 + 28b + 2b2 = 1225 + -1225 Combine like terms: 1225 + -1225 = 0 -1029 + 28b + 2b2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -514.5 + 14b + b2 = 0 Move the constant term to the right: Add '514.5' to each side of the equation. -514.5 + 14b + 514.5 + b2 = 0 + 514.5 Reorder the terms: -514.5 + 514.5 + 14b + b2 = 0 + 514.5 Combine like terms: -514.5 + 514.5 = 0.0 0.0 + 14b + b2 = 0 + 514.5 14b + b2 = 0 + 514.5 Combine like terms: 0 + 514.5 = 514.5 14b + b2 = 514.5 The b term is 14b. Take half its coefficient (7). Square it (49) and add it to both sides. Add '49' to each side of the equation. 14b + 49 + b2 = 514.5 + 49 Reorder the terms: 49 + 14b + b2 = 514.5 + 49 Combine like terms: 514.5 + 49 = 563.5 49 + 14b + b2 = 563.5 Factor a perfect square on the left side: (b + 7)(b + 7) = 563.5 Calculate the square root of the right side: 23.738154941 Break this problem into two subproblems by setting (b + 7) equal to 23.738154941 and -23.738154941.Subproblem 1
b + 7 = 23.738154941 Simplifying b + 7 = 23.738154941 Reorder the terms: 7 + b = 23.738154941 Solving 7 + b = 23.738154941 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-7' to each side of the equation. 7 + -7 + b = 23.738154941 + -7 Combine like terms: 7 + -7 = 0 0 + b = 23.738154941 + -7 b = 23.738154941 + -7 Combine like terms: 23.738154941 + -7 = 16.738154941 b = 16.738154941 Simplifying b = 16.738154941Subproblem 2
b + 7 = -23.738154941 Simplifying b + 7 = -23.738154941 Reorder the terms: 7 + b = -23.738154941 Solving 7 + b = -23.738154941 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-7' to each side of the equation. 7 + -7 + b = -23.738154941 + -7 Combine like terms: 7 + -7 = 0 0 + b = -23.738154941 + -7 b = -23.738154941 + -7 Combine like terms: -23.738154941 + -7 = -30.738154941 b = -30.738154941 Simplifying b = -30.738154941Solution
The solution to the problem is based on the solutions from the subproblems. b = {16.738154941, -30.738154941}
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