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Simplifying 6b2 + 16b + -9 = 0 Reorder the terms: -9 + 16b + 6b2 = 0 Solving -9 + 16b + 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'. -1.5 + 2.666666667b + b2 = 0 Move the constant term to the right: Add '1.5' to each side of the equation. -1.5 + 2.666666667b + 1.5 + b2 = 0 + 1.5 Reorder the terms: -1.5 + 1.5 + 2.666666667b + b2 = 0 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + 2.666666667b + b2 = 0 + 1.5 2.666666667b + b2 = 0 + 1.5 Combine like terms: 0 + 1.5 = 1.5 2.666666667b + b2 = 1.5 The b term is 2.666666667b. Take half its coefficient (1.333333334). Square it (1.777777780) and add it to both sides. Add '1.777777780' to each side of the equation. 2.666666667b + 1.777777780 + b2 = 1.5 + 1.777777780 Reorder the terms: 1.777777780 + 2.666666667b + b2 = 1.5 + 1.777777780 Combine like terms: 1.5 + 1.777777780 = 3.27777778 1.777777780 + 2.666666667b + b2 = 3.27777778 Factor a perfect square on the left side: (b + 1.333333334)(b + 1.333333334) = 3.27777778 Calculate the square root of the right side: 1.810463416 Break this problem into two subproblems by setting (b + 1.333333334) equal to 1.810463416 and -1.810463416.Subproblem 1
b + 1.333333334 = 1.810463416 Simplifying b + 1.333333334 = 1.810463416 Reorder the terms: 1.333333334 + b = 1.810463416 Solving 1.333333334 + b = 1.810463416 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-1.333333334' to each side of the equation. 1.333333334 + -1.333333334 + b = 1.810463416 + -1.333333334 Combine like terms: 1.333333334 + -1.333333334 = 0.000000000 0.000000000 + b = 1.810463416 + -1.333333334 b = 1.810463416 + -1.333333334 Combine like terms: 1.810463416 + -1.333333334 = 0.477130082 b = 0.477130082 Simplifying b = 0.477130082Subproblem 2
b + 1.333333334 = -1.810463416 Simplifying b + 1.333333334 = -1.810463416 Reorder the terms: 1.333333334 + b = -1.810463416 Solving 1.333333334 + b = -1.810463416 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-1.333333334' to each side of the equation. 1.333333334 + -1.333333334 + b = -1.810463416 + -1.333333334 Combine like terms: 1.333333334 + -1.333333334 = 0.000000000 0.000000000 + b = -1.810463416 + -1.333333334 b = -1.810463416 + -1.333333334 Combine like terms: -1.810463416 + -1.333333334 = -3.14379675 b = -3.14379675 Simplifying b = -3.14379675Solution
The solution to the problem is based on the solutions from the subproblems. b = {0.477130082, -3.14379675}
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