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Simplifying -7(1 + -3b) + b(8b + -6) = 0 (1 * -7 + -3b * -7) + b(8b + -6) = 0 (-7 + 21b) + b(8b + -6) = 0 Reorder the terms: -7 + 21b + b(-6 + 8b) = 0 -7 + 21b + (-6 * b + 8b * b) = 0 -7 + 21b + (-6b + 8b2) = 0 Combine like terms: 21b + -6b = 15b -7 + 15b + 8b2 = 0 Solving -7 + 15b + 8b2 = 0 Solving for variable 'b'. Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -0.875 + 1.875b + b2 = 0 Move the constant term to the right: Add '0.875' to each side of the equation. -0.875 + 1.875b + 0.875 + b2 = 0 + 0.875 Reorder the terms: -0.875 + 0.875 + 1.875b + b2 = 0 + 0.875 Combine like terms: -0.875 + 0.875 = 0.000 0.000 + 1.875b + b2 = 0 + 0.875 1.875b + b2 = 0 + 0.875 Combine like terms: 0 + 0.875 = 0.875 1.875b + b2 = 0.875 The b term is 1.875b. Take half its coefficient (0.9375). Square it (0.87890625) and add it to both sides. Add '0.87890625' to each side of the equation. 1.875b + 0.87890625 + b2 = 0.875 + 0.87890625 Reorder the terms: 0.87890625 + 1.875b + b2 = 0.875 + 0.87890625 Combine like terms: 0.875 + 0.87890625 = 1.75390625 0.87890625 + 1.875b + b2 = 1.75390625 Factor a perfect square on the left side: (b + 0.9375)(b + 0.9375) = 1.75390625 Calculate the square root of the right side: 1.324351256 Break this problem into two subproblems by setting (b + 0.9375) equal to 1.324351256 and -1.324351256.Subproblem 1
b + 0.9375 = 1.324351256 Simplifying b + 0.9375 = 1.324351256 Reorder the terms: 0.9375 + b = 1.324351256 Solving 0.9375 + b = 1.324351256 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-0.9375' to each side of the equation. 0.9375 + -0.9375 + b = 1.324351256 + -0.9375 Combine like terms: 0.9375 + -0.9375 = 0.0000 0.0000 + b = 1.324351256 + -0.9375 b = 1.324351256 + -0.9375 Combine like terms: 1.324351256 + -0.9375 = 0.386851256 b = 0.386851256 Simplifying b = 0.386851256Subproblem 2
b + 0.9375 = -1.324351256 Simplifying b + 0.9375 = -1.324351256 Reorder the terms: 0.9375 + b = -1.324351256 Solving 0.9375 + b = -1.324351256 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-0.9375' to each side of the equation. 0.9375 + -0.9375 + b = -1.324351256 + -0.9375 Combine like terms: 0.9375 + -0.9375 = 0.0000 0.0000 + b = -1.324351256 + -0.9375 b = -1.324351256 + -0.9375 Combine like terms: -1.324351256 + -0.9375 = -2.261851256 b = -2.261851256 Simplifying b = -2.261851256Solution
The solution to the problem is based on the solutions from the subproblems. b = {0.386851256, -2.261851256}
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