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Simplifying 3b2 + 12b + 1 = 3 Reorder the terms: 1 + 12b + 3b2 = 3 Solving 1 + 12b + 3b2 = 3 Solving for variable 'b'. Reorder the terms: 1 + -3 + 12b + 3b2 = 3 + -3 Combine like terms: 1 + -3 = -2 -2 + 12b + 3b2 = 3 + -3 Combine like terms: 3 + -3 = 0 -2 + 12b + 3b2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -0.6666666667 + 4b + b2 = 0 Move the constant term to the right: Add '0.6666666667' to each side of the equation. -0.6666666667 + 4b + 0.6666666667 + b2 = 0 + 0.6666666667 Reorder the terms: -0.6666666667 + 0.6666666667 + 4b + b2 = 0 + 0.6666666667 Combine like terms: -0.6666666667 + 0.6666666667 = 0.0000000000 0.0000000000 + 4b + b2 = 0 + 0.6666666667 4b + b2 = 0 + 0.6666666667 Combine like terms: 0 + 0.6666666667 = 0.6666666667 4b + b2 = 0.6666666667 The b term is 4b. Take half its coefficient (2). Square it (4) and add it to both sides. Add '4' to each side of the equation. 4b + 4 + b2 = 0.6666666667 + 4 Reorder the terms: 4 + 4b + b2 = 0.6666666667 + 4 Combine like terms: 0.6666666667 + 4 = 4.6666666667 4 + 4b + b2 = 4.6666666667 Factor a perfect square on the left side: (b + 2)(b + 2) = 4.6666666667 Calculate the square root of the right side: 2.160246899 Break this problem into two subproblems by setting (b + 2) equal to 2.160246899 and -2.160246899.Subproblem 1
b + 2 = 2.160246899 Simplifying b + 2 = 2.160246899 Reorder the terms: 2 + b = 2.160246899 Solving 2 + b = 2.160246899 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + b = 2.160246899 + -2 Combine like terms: 2 + -2 = 0 0 + b = 2.160246899 + -2 b = 2.160246899 + -2 Combine like terms: 2.160246899 + -2 = 0.160246899 b = 0.160246899 Simplifying b = 0.160246899Subproblem 2
b + 2 = -2.160246899 Simplifying b + 2 = -2.160246899 Reorder the terms: 2 + b = -2.160246899 Solving 2 + b = -2.160246899 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-2' to each side of the equation. 2 + -2 + b = -2.160246899 + -2 Combine like terms: 2 + -2 = 0 0 + b = -2.160246899 + -2 b = -2.160246899 + -2 Combine like terms: -2.160246899 + -2 = -4.160246899 b = -4.160246899 Simplifying b = -4.160246899Solution
The solution to the problem is based on the solutions from the subproblems. b = {0.160246899, -4.160246899}
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