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Simplifying 6b(b + -5) + -5b = -3 Reorder the terms: 6b(-5 + b) + -5b = -3 (-5 * 6b + b * 6b) + -5b = -3 (-30b + 6b2) + -5b = -3 Reorder the terms: -30b + -5b + 6b2 = -3 Combine like terms: -30b + -5b = -35b -35b + 6b2 = -3 Solving -35b + 6b2 = -3 Solving for variable 'b'. Reorder the terms: 3 + -35b + 6b2 = -3 + 3 Combine like terms: -3 + 3 = 0 3 + -35b + 6b2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. 0.5 + -5.833333333b + b2 = 0 Move the constant term to the right: Add '-0.5' to each side of the equation. 0.5 + -5.833333333b + -0.5 + b2 = 0 + -0.5 Reorder the terms: 0.5 + -0.5 + -5.833333333b + b2 = 0 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + -5.833333333b + b2 = 0 + -0.5 -5.833333333b + b2 = 0 + -0.5 Combine like terms: 0 + -0.5 = -0.5 -5.833333333b + b2 = -0.5 The b term is -5.833333333b. Take half its coefficient (-2.916666667). Square it (8.506944446) and add it to both sides. Add '8.506944446' to each side of the equation. -5.833333333b + 8.506944446 + b2 = -0.5 + 8.506944446 Reorder the terms: 8.506944446 + -5.833333333b + b2 = -0.5 + 8.506944446 Combine like terms: -0.5 + 8.506944446 = 8.006944446 8.506944446 + -5.833333333b + b2 = 8.006944446 Factor a perfect square on the left side: (b + -2.916666667)(b + -2.916666667) = 8.006944446 Calculate the square root of the right side: 2.829654475 Break this problem into two subproblems by setting (b + -2.916666667) equal to 2.829654475 and -2.829654475.Subproblem 1
b + -2.916666667 = 2.829654475 Simplifying b + -2.916666667 = 2.829654475 Reorder the terms: -2.916666667 + b = 2.829654475 Solving -2.916666667 + b = 2.829654475 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '2.916666667' to each side of the equation. -2.916666667 + 2.916666667 + b = 2.829654475 + 2.916666667 Combine like terms: -2.916666667 + 2.916666667 = 0.000000000 0.000000000 + b = 2.829654475 + 2.916666667 b = 2.829654475 + 2.916666667 Combine like terms: 2.829654475 + 2.916666667 = 5.746321142 b = 5.746321142 Simplifying b = 5.746321142Subproblem 2
b + -2.916666667 = -2.829654475 Simplifying b + -2.916666667 = -2.829654475 Reorder the terms: -2.916666667 + b = -2.829654475 Solving -2.916666667 + b = -2.829654475 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '2.916666667' to each side of the equation. -2.916666667 + 2.916666667 + b = -2.829654475 + 2.916666667 Combine like terms: -2.916666667 + 2.916666667 = 0.000000000 0.000000000 + b = -2.829654475 + 2.916666667 b = -2.829654475 + 2.916666667 Combine like terms: -2.829654475 + 2.916666667 = 0.087012192 b = 0.087012192 Simplifying b = 0.087012192Solution
The solution to the problem is based on the solutions from the subproblems. b = {5.746321142, 0.087012192}
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