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Simplifying 7b(8b + -9) + -1(b + -9) = 0 Reorder the terms: 7b(-9 + 8b) + -1(b + -9) = 0 (-9 * 7b + 8b * 7b) + -1(b + -9) = 0 (-63b + 56b2) + -1(b + -9) = 0 Reorder the terms: -63b + 56b2 + -1(-9 + b) = 0 -63b + 56b2 + (-9 * -1 + b * -1) = 0 -63b + 56b2 + (9 + -1b) = 0 Reorder the terms: 9 + -63b + -1b + 56b2 = 0 Combine like terms: -63b + -1b = -64b 9 + -64b + 56b2 = 0 Solving 9 + -64b + 56b2 = 0 Solving for variable 'b'. Begin completing the square. Divide all terms by 56 the coefficient of the squared term: Divide each side by '56'. 0.1607142857 + -1.142857143b + b2 = 0 Move the constant term to the right: Add '-0.1607142857' to each side of the equation. 0.1607142857 + -1.142857143b + -0.1607142857 + b2 = 0 + -0.1607142857 Reorder the terms: 0.1607142857 + -0.1607142857 + -1.142857143b + b2 = 0 + -0.1607142857 Combine like terms: 0.1607142857 + -0.1607142857 = 0.0000000000 0.0000000000 + -1.142857143b + b2 = 0 + -0.1607142857 -1.142857143b + b2 = 0 + -0.1607142857 Combine like terms: 0 + -0.1607142857 = -0.1607142857 -1.142857143b + b2 = -0.1607142857 The b term is -1.142857143b. Take half its coefficient (-0.5714285715). Square it (0.3265306123) and add it to both sides. Add '0.3265306123' to each side of the equation. -1.142857143b + 0.3265306123 + b2 = -0.1607142857 + 0.3265306123 Reorder the terms: 0.3265306123 + -1.142857143b + b2 = -0.1607142857 + 0.3265306123 Combine like terms: -0.1607142857 + 0.3265306123 = 0.1658163266 0.3265306123 + -1.142857143b + b2 = 0.1658163266 Factor a perfect square on the left side: (b + -0.5714285715)(b + -0.5714285715) = 0.1658163266 Calculate the square root of the right side: 0.407205509 Break this problem into two subproblems by setting (b + -0.5714285715) equal to 0.407205509 and -0.407205509.Subproblem 1
b + -0.5714285715 = 0.407205509 Simplifying b + -0.5714285715 = 0.407205509 Reorder the terms: -0.5714285715 + b = 0.407205509 Solving -0.5714285715 + b = 0.407205509 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '0.5714285715' to each side of the equation. -0.5714285715 + 0.5714285715 + b = 0.407205509 + 0.5714285715 Combine like terms: -0.5714285715 + 0.5714285715 = 0.0000000000 0.0000000000 + b = 0.407205509 + 0.5714285715 b = 0.407205509 + 0.5714285715 Combine like terms: 0.407205509 + 0.5714285715 = 0.9786340805 b = 0.9786340805 Simplifying b = 0.9786340805Subproblem 2
b + -0.5714285715 = -0.407205509 Simplifying b + -0.5714285715 = -0.407205509 Reorder the terms: -0.5714285715 + b = -0.407205509 Solving -0.5714285715 + b = -0.407205509 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '0.5714285715' to each side of the equation. -0.5714285715 + 0.5714285715 + b = -0.407205509 + 0.5714285715 Combine like terms: -0.5714285715 + 0.5714285715 = 0.0000000000 0.0000000000 + b = -0.407205509 + 0.5714285715 b = -0.407205509 + 0.5714285715 Combine like terms: -0.407205509 + 0.5714285715 = 0.1642230625 b = 0.1642230625 Simplifying b = 0.1642230625Solution
The solution to the problem is based on the solutions from the subproblems. b = {0.9786340805, 0.1642230625}
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