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Simplifying 6b2 + -13b + 3 = 0 Reorder the terms: 3 + -13b + 6b2 = 0 Solving 3 + -13b + 6b2 = 0 Solving for variable 'b'. Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. 0.5 + -2.166666667b + b2 = 0 Move the constant term to the right: Add '-0.5' to each side of the equation. 0.5 + -2.166666667b + -0.5 + b2 = 0 + -0.5 Reorder the terms: 0.5 + -0.5 + -2.166666667b + b2 = 0 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + -2.166666667b + b2 = 0 + -0.5 -2.166666667b + b2 = 0 + -0.5 Combine like terms: 0 + -0.5 = -0.5 -2.166666667b + b2 = -0.5 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 = -0.5 + 1.173611113 Reorder the terms: 1.173611113 + -2.166666667b + b2 = -0.5 + 1.173611113 Combine like terms: -0.5 + 1.173611113 = 0.673611113 1.173611113 + -2.166666667b + b2 = 0.673611113 Factor a perfect square on the left side: (b + -1.083333334)(b + -1.083333334) = 0.673611113 Calculate the square root of the right side: 0.820738151 Break this problem into two subproblems by setting (b + -1.083333334) equal to 0.820738151 and -0.820738151.Subproblem 1
b + -1.083333334 = 0.820738151 Simplifying b + -1.083333334 = 0.820738151 Reorder the terms: -1.083333334 + b = 0.820738151 Solving -1.083333334 + b = 0.820738151 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 = 0.820738151 + 1.083333334 Combine like terms: -1.083333334 + 1.083333334 = 0.000000000 0.000000000 + b = 0.820738151 + 1.083333334 b = 0.820738151 + 1.083333334 Combine like terms: 0.820738151 + 1.083333334 = 1.904071485 b = 1.904071485 Simplifying b = 1.904071485Subproblem 2
b + -1.083333334 = -0.820738151 Simplifying b + -1.083333334 = -0.820738151 Reorder the terms: -1.083333334 + b = -0.820738151 Solving -1.083333334 + b = -0.820738151 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 = -0.820738151 + 1.083333334 Combine like terms: -1.083333334 + 1.083333334 = 0.000000000 0.000000000 + b = -0.820738151 + 1.083333334 b = -0.820738151 + 1.083333334 Combine like terms: -0.820738151 + 1.083333334 = 0.262595183 b = 0.262595183 Simplifying b = 0.262595183Solution
The solution to the problem is based on the solutions from the subproblems. b = {1.904071485, 0.262595183}
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