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Simplifying 3b2 + -12b = -4 Reorder the terms: -12b + 3b2 = -4 Solving -12b + 3b2 = -4 Solving for variable 'b'. Reorder the terms: 4 + -12b + 3b2 = -4 + 4 Combine like terms: -4 + 4 = 0 4 + -12b + 3b2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. 1.333333333 + -4b + b2 = 0 Move the constant term to the right: Add '-1.333333333' to each side of the equation. 1.333333333 + -4b + -1.333333333 + b2 = 0 + -1.333333333 Reorder the terms: 1.333333333 + -1.333333333 + -4b + b2 = 0 + -1.333333333 Combine like terms: 1.333333333 + -1.333333333 = 0.000000000 0.000000000 + -4b + b2 = 0 + -1.333333333 -4b + b2 = 0 + -1.333333333 Combine like terms: 0 + -1.333333333 = -1.333333333 -4b + b2 = -1.333333333 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 = -1.333333333 + 4 Reorder the terms: 4 + -4b + b2 = -1.333333333 + 4 Combine like terms: -1.333333333 + 4 = 2.666666667 4 + -4b + b2 = 2.666666667 Factor a perfect square on the left side: (b + -2)(b + -2) = 2.666666667 Calculate the square root of the right side: 1.632993162 Break this problem into two subproblems by setting (b + -2) equal to 1.632993162 and -1.632993162.Subproblem 1
b + -2 = 1.632993162 Simplifying b + -2 = 1.632993162 Reorder the terms: -2 + b = 1.632993162 Solving -2 + b = 1.632993162 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 = 1.632993162 + 2 Combine like terms: -2 + 2 = 0 0 + b = 1.632993162 + 2 b = 1.632993162 + 2 Combine like terms: 1.632993162 + 2 = 3.632993162 b = 3.632993162 Simplifying b = 3.632993162Subproblem 2
b + -2 = -1.632993162 Simplifying b + -2 = -1.632993162 Reorder the terms: -2 + b = -1.632993162 Solving -2 + b = -1.632993162 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 = -1.632993162 + 2 Combine like terms: -2 + 2 = 0 0 + b = -1.632993162 + 2 b = -1.632993162 + 2 Combine like terms: -1.632993162 + 2 = 0.367006838 b = 0.367006838 Simplifying b = 0.367006838Solution
The solution to the problem is based on the solutions from the subproblems. b = {3.632993162, 0.367006838}
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