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Simplifying 3b2 + -6b + 2 = 0 Reorder the terms: 2 + -6b + 3b2 = 0 Solving 2 + -6b + 3b2 = 0 Solving for variable 'b'. Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. 0.6666666667 + -2b + b2 = 0 Move the constant term to the right: Add '-0.6666666667' to each side of the equation. 0.6666666667 + -2b + -0.6666666667 + b2 = 0 + -0.6666666667 Reorder the terms: 0.6666666667 + -0.6666666667 + -2b + b2 = 0 + -0.6666666667 Combine like terms: 0.6666666667 + -0.6666666667 = 0.0000000000 0.0000000000 + -2b + b2 = 0 + -0.6666666667 -2b + b2 = 0 + -0.6666666667 Combine like terms: 0 + -0.6666666667 = -0.6666666667 -2b + b2 = -0.6666666667 The b term is -2b. Take half its coefficient (-1). Square it (1) and add it to both sides. Add '1' to each side of the equation. -2b + 1 + b2 = -0.6666666667 + 1 Reorder the terms: 1 + -2b + b2 = -0.6666666667 + 1 Combine like terms: -0.6666666667 + 1 = 0.3333333333 1 + -2b + b2 = 0.3333333333 Factor a perfect square on the left side: (b + -1)(b + -1) = 0.3333333333 Calculate the square root of the right side: 0.577350269 Break this problem into two subproblems by setting (b + -1) equal to 0.577350269 and -0.577350269.Subproblem 1
b + -1 = 0.577350269 Simplifying b + -1 = 0.577350269 Reorder the terms: -1 + b = 0.577350269 Solving -1 + b = 0.577350269 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + b = 0.577350269 + 1 Combine like terms: -1 + 1 = 0 0 + b = 0.577350269 + 1 b = 0.577350269 + 1 Combine like terms: 0.577350269 + 1 = 1.577350269 b = 1.577350269 Simplifying b = 1.577350269Subproblem 2
b + -1 = -0.577350269 Simplifying b + -1 = -0.577350269 Reorder the terms: -1 + b = -0.577350269 Solving -1 + b = -0.577350269 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '1' to each side of the equation. -1 + 1 + b = -0.577350269 + 1 Combine like terms: -1 + 1 = 0 0 + b = -0.577350269 + 1 b = -0.577350269 + 1 Combine like terms: -0.577350269 + 1 = 0.422649731 b = 0.422649731 Simplifying b = 0.422649731Solution
The solution to the problem is based on the solutions from the subproblems. b = {1.577350269, 0.422649731}
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