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Simplifying 2b + 3b(3b + 1) = 17 Reorder the terms: 2b + 3b(1 + 3b) = 17 2b + (1 * 3b + 3b * 3b) = 17 2b + (3b + 9b2) = 17 Combine like terms: 2b + 3b = 5b 5b + 9b2 = 17 Solving 5b + 9b2 = 17 Solving for variable 'b'. Reorder the terms: -17 + 5b + 9b2 = 17 + -17 Combine like terms: 17 + -17 = 0 -17 + 5b + 9b2 = 0 Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. -1.888888889 + 0.5555555556b + b2 = 0 Move the constant term to the right: Add '1.888888889' to each side of the equation. -1.888888889 + 0.5555555556b + 1.888888889 + b2 = 0 + 1.888888889 Reorder the terms: -1.888888889 + 1.888888889 + 0.5555555556b + b2 = 0 + 1.888888889 Combine like terms: -1.888888889 + 1.888888889 = 0.000000000 0.000000000 + 0.5555555556b + b2 = 0 + 1.888888889 0.5555555556b + b2 = 0 + 1.888888889 Combine like terms: 0 + 1.888888889 = 1.888888889 0.5555555556b + b2 = 1.888888889 The b term is 0.5555555556b. Take half its coefficient (0.2777777778). Square it (0.07716049384) and add it to both sides. Add '0.07716049384' to each side of the equation. 0.5555555556b + 0.07716049384 + b2 = 1.888888889 + 0.07716049384 Reorder the terms: 0.07716049384 + 0.5555555556b + b2 = 1.888888889 + 0.07716049384 Combine like terms: 1.888888889 + 0.07716049384 = 1.96604938284 0.07716049384 + 0.5555555556b + b2 = 1.96604938284 Factor a perfect square on the left side: (b + 0.2777777778)(b + 0.2777777778) = 1.96604938284 Calculate the square root of the right side: 1.402158829 Break this problem into two subproblems by setting (b + 0.2777777778) equal to 1.402158829 and -1.402158829.Subproblem 1
b + 0.2777777778 = 1.402158829 Simplifying b + 0.2777777778 = 1.402158829 Reorder the terms: 0.2777777778 + b = 1.402158829 Solving 0.2777777778 + b = 1.402158829 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-0.2777777778' to each side of the equation. 0.2777777778 + -0.2777777778 + b = 1.402158829 + -0.2777777778 Combine like terms: 0.2777777778 + -0.2777777778 = 0.0000000000 0.0000000000 + b = 1.402158829 + -0.2777777778 b = 1.402158829 + -0.2777777778 Combine like terms: 1.402158829 + -0.2777777778 = 1.1243810512 b = 1.1243810512 Simplifying b = 1.1243810512Subproblem 2
b + 0.2777777778 = -1.402158829 Simplifying b + 0.2777777778 = -1.402158829 Reorder the terms: 0.2777777778 + b = -1.402158829 Solving 0.2777777778 + b = -1.402158829 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-0.2777777778' to each side of the equation. 0.2777777778 + -0.2777777778 + b = -1.402158829 + -0.2777777778 Combine like terms: 0.2777777778 + -0.2777777778 = 0.0000000000 0.0000000000 + b = -1.402158829 + -0.2777777778 b = -1.402158829 + -0.2777777778 Combine like terms: -1.402158829 + -0.2777777778 = -1.6799366068 b = -1.6799366068 Simplifying b = -1.6799366068Solution
The solution to the problem is based on the solutions from the subproblems. b = {1.1243810512, -1.6799366068}
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