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Simplifying 9(2b + 1) * 2b = 29 Reorder the terms: 9(1 + 2b) * 2b = 29 Reorder the terms for easier multiplication: 9 * 2b(1 + 2b) = 29 Multiply 9 * 2 18b(1 + 2b) = 29 (1 * 18b + 2b * 18b) = 29 (18b + 36b2) = 29 Solving 18b + 36b2 = 29 Solving for variable 'b'. Reorder the terms: -29 + 18b + 36b2 = 29 + -29 Combine like terms: 29 + -29 = 0 -29 + 18b + 36b2 = 0 Begin completing the square. Divide all terms by 36 the coefficient of the squared term: Divide each side by '36'. -0.8055555556 + 0.5b + b2 = 0 Move the constant term to the right: Add '0.8055555556' to each side of the equation. -0.8055555556 + 0.5b + 0.8055555556 + b2 = 0 + 0.8055555556 Reorder the terms: -0.8055555556 + 0.8055555556 + 0.5b + b2 = 0 + 0.8055555556 Combine like terms: -0.8055555556 + 0.8055555556 = 0.0000000000 0.0000000000 + 0.5b + b2 = 0 + 0.8055555556 0.5b + b2 = 0 + 0.8055555556 Combine like terms: 0 + 0.8055555556 = 0.8055555556 0.5b + b2 = 0.8055555556 The b term is 0.5b. Take half its coefficient (0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. 0.5b + 0.0625 + b2 = 0.8055555556 + 0.0625 Reorder the terms: 0.0625 + 0.5b + b2 = 0.8055555556 + 0.0625 Combine like terms: 0.8055555556 + 0.0625 = 0.8680555556 0.0625 + 0.5b + b2 = 0.8680555556 Factor a perfect square on the left side: (b + 0.25)(b + 0.25) = 0.8680555556 Calculate the square root of the right side: 0.931694991 Break this problem into two subproblems by setting (b + 0.25) equal to 0.931694991 and -0.931694991.Subproblem 1
b + 0.25 = 0.931694991 Simplifying b + 0.25 = 0.931694991 Reorder the terms: 0.25 + b = 0.931694991 Solving 0.25 + b = 0.931694991 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + b = 0.931694991 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + b = 0.931694991 + -0.25 b = 0.931694991 + -0.25 Combine like terms: 0.931694991 + -0.25 = 0.681694991 b = 0.681694991 Simplifying b = 0.681694991Subproblem 2
b + 0.25 = -0.931694991 Simplifying b + 0.25 = -0.931694991 Reorder the terms: 0.25 + b = -0.931694991 Solving 0.25 + b = -0.931694991 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + b = -0.931694991 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + b = -0.931694991 + -0.25 b = -0.931694991 + -0.25 Combine like terms: -0.931694991 + -0.25 = -1.181694991 b = -1.181694991 Simplifying b = -1.181694991Solution
The solution to the problem is based on the solutions from the subproblems. b = {0.681694991, -1.181694991}
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