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Simplifying 5 + -7b = 2(b + 4) * b Reorder the terms: 5 + -7b = 2(4 + b) * b Reorder the terms for easier multiplication: 5 + -7b = 2b(4 + b) 5 + -7b = (4 * 2b + b * 2b) 5 + -7b = (8b + 2b2) Solving 5 + -7b = 8b + 2b2 Solving for variable 'b'. Combine like terms: -7b + -8b = -15b 5 + -15b + -2b2 = 8b + 2b2 + -8b + -2b2 Reorder the terms: 5 + -15b + -2b2 = 8b + -8b + 2b2 + -2b2 Combine like terms: 8b + -8b = 0 5 + -15b + -2b2 = 0 + 2b2 + -2b2 5 + -15b + -2b2 = 2b2 + -2b2 Combine like terms: 2b2 + -2b2 = 0 5 + -15b + -2b2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -2.5 + 7.5b + b2 = 0 Move the constant term to the right: Add '2.5' to each side of the equation. -2.5 + 7.5b + 2.5 + b2 = 0 + 2.5 Reorder the terms: -2.5 + 2.5 + 7.5b + b2 = 0 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + 7.5b + b2 = 0 + 2.5 7.5b + b2 = 0 + 2.5 Combine like terms: 0 + 2.5 = 2.5 7.5b + b2 = 2.5 The b term is 7.5b. Take half its coefficient (3.75). Square it (14.0625) and add it to both sides. Add '14.0625' to each side of the equation. 7.5b + 14.0625 + b2 = 2.5 + 14.0625 Reorder the terms: 14.0625 + 7.5b + b2 = 2.5 + 14.0625 Combine like terms: 2.5 + 14.0625 = 16.5625 14.0625 + 7.5b + b2 = 16.5625 Factor a perfect square on the left side: (b + 3.75)(b + 3.75) = 16.5625 Calculate the square root of the right side: 4.069705149 Break this problem into two subproblems by setting (b + 3.75) equal to 4.069705149 and -4.069705149.Subproblem 1
b + 3.75 = 4.069705149 Simplifying b + 3.75 = 4.069705149 Reorder the terms: 3.75 + b = 4.069705149 Solving 3.75 + b = 4.069705149 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-3.75' to each side of the equation. 3.75 + -3.75 + b = 4.069705149 + -3.75 Combine like terms: 3.75 + -3.75 = 0.00 0.00 + b = 4.069705149 + -3.75 b = 4.069705149 + -3.75 Combine like terms: 4.069705149 + -3.75 = 0.319705149 b = 0.319705149 Simplifying b = 0.319705149Subproblem 2
b + 3.75 = -4.069705149 Simplifying b + 3.75 = -4.069705149 Reorder the terms: 3.75 + b = -4.069705149 Solving 3.75 + b = -4.069705149 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-3.75' to each side of the equation. 3.75 + -3.75 + b = -4.069705149 + -3.75 Combine like terms: 3.75 + -3.75 = 0.00 0.00 + b = -4.069705149 + -3.75 b = -4.069705149 + -3.75 Combine like terms: -4.069705149 + -3.75 = -7.819705149 b = -7.819705149 Simplifying b = -7.819705149Solution
The solution to the problem is based on the solutions from the subproblems. b = {0.319705149, -7.819705149}
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