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Simplifying -6b2 + -13b + 3 = -3 Reorder the terms: 3 + -13b + -6b2 = -3 Solving 3 + -13b + -6b2 = -3 Solving for variable 'b'. Reorder the terms: 3 + 3 + -13b + -6b2 = -3 + 3 Combine like terms: 3 + 3 = 6 6 + -13b + -6b2 = -3 + 3 Combine like terms: -3 + 3 = 0 6 + -13b + -6b2 = 0 Begin completing the square. Divide all terms by -6 the coefficient of the squared term: Divide each side by '-6'. -1 + 2.166666667b + b2 = 0 Move the constant term to the right: Add '1' to each side of the equation. -1 + 2.166666667b + 1 + b2 = 0 + 1 Reorder the terms: -1 + 1 + 2.166666667b + b2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + 2.166666667b + b2 = 0 + 1 2.166666667b + b2 = 0 + 1 Combine like terms: 0 + 1 = 1 2.166666667b + b2 = 1 The b term is 2.166666667b. Take half its coefficient (1.083333334). Square it (1.173611113) and add it to both sides. Add '1.173611113' to each side of the equation. 2.166666667b + 1.173611113 + b2 = 1 + 1.173611113 Reorder the terms: 1.173611113 + 2.166666667b + b2 = 1 + 1.173611113 Combine like terms: 1 + 1.173611113 = 2.173611113 1.173611113 + 2.166666667b + b2 = 2.173611113 Factor a perfect square on the left side: (b + 1.083333334)(b + 1.083333334) = 2.173611113 Calculate the square root of the right side: 1.474317168 Break this problem into two subproblems by setting (b + 1.083333334) equal to 1.474317168 and -1.474317168.Subproblem 1
b + 1.083333334 = 1.474317168 Simplifying b + 1.083333334 = 1.474317168 Reorder the terms: 1.083333334 + b = 1.474317168 Solving 1.083333334 + b = 1.474317168 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-1.083333334' to each side of the equation. 1.083333334 + -1.083333334 + b = 1.474317168 + -1.083333334 Combine like terms: 1.083333334 + -1.083333334 = 0.000000000 0.000000000 + b = 1.474317168 + -1.083333334 b = 1.474317168 + -1.083333334 Combine like terms: 1.474317168 + -1.083333334 = 0.390983834 b = 0.390983834 Simplifying b = 0.390983834Subproblem 2
b + 1.083333334 = -1.474317168 Simplifying b + 1.083333334 = -1.474317168 Reorder the terms: 1.083333334 + b = -1.474317168 Solving 1.083333334 + b = -1.474317168 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-1.083333334' to each side of the equation. 1.083333334 + -1.083333334 + b = -1.474317168 + -1.083333334 Combine like terms: 1.083333334 + -1.083333334 = 0.000000000 0.000000000 + b = -1.474317168 + -1.083333334 b = -1.474317168 + -1.083333334 Combine like terms: -1.474317168 + -1.083333334 = -2.557650502 b = -2.557650502 Simplifying b = -2.557650502Solution
The solution to the problem is based on the solutions from the subproblems. b = {0.390983834, -2.557650502}
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