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Simplifying (4b2 + -3b + 7) + (-8b2 + -5b + 16) = 0 Reorder the terms: (7 + -3b + 4b2) + (-8b2 + -5b + 16) = 0 Remove parenthesis around (7 + -3b + 4b2) 7 + -3b + 4b2 + (-8b2 + -5b + 16) = 0 Reorder the terms: 7 + -3b + 4b2 + (16 + -5b + -8b2) = 0 Remove parenthesis around (16 + -5b + -8b2) 7 + -3b + 4b2 + 16 + -5b + -8b2 = 0 Reorder the terms: 7 + 16 + -3b + -5b + 4b2 + -8b2 = 0 Combine like terms: 7 + 16 = 23 23 + -3b + -5b + 4b2 + -8b2 = 0 Combine like terms: -3b + -5b = -8b 23 + -8b + 4b2 + -8b2 = 0 Combine like terms: 4b2 + -8b2 = -4b2 23 + -8b + -4b2 = 0 Solving 23 + -8b + -4b2 = 0 Solving for variable 'b'. Begin completing the square. Divide all terms by -4 the coefficient of the squared term: Divide each side by '-4'. -5.75 + 2b + b2 = 0 Move the constant term to the right: Add '5.75' to each side of the equation. -5.75 + 2b + 5.75 + b2 = 0 + 5.75 Reorder the terms: -5.75 + 5.75 + 2b + b2 = 0 + 5.75 Combine like terms: -5.75 + 5.75 = 0.00 0.00 + 2b + b2 = 0 + 5.75 2b + b2 = 0 + 5.75 Combine like terms: 0 + 5.75 = 5.75 2b + b2 = 5.75 The b term is 2b. Take half its coefficient (1). Square it (1) and add it to both sides. Add '1' to each side of the equation. 2b + 1 + b2 = 5.75 + 1 Reorder the terms: 1 + 2b + b2 = 5.75 + 1 Combine like terms: 5.75 + 1 = 6.75 1 + 2b + b2 = 6.75 Factor a perfect square on the left side: (b + 1)(b + 1) = 6.75 Calculate the square root of the right side: 2.598076211 Break this problem into two subproblems by setting (b + 1) equal to 2.598076211 and -2.598076211.Subproblem 1
b + 1 = 2.598076211 Simplifying b + 1 = 2.598076211 Reorder the terms: 1 + b = 2.598076211 Solving 1 + b = 2.598076211 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + b = 2.598076211 + -1 Combine like terms: 1 + -1 = 0 0 + b = 2.598076211 + -1 b = 2.598076211 + -1 Combine like terms: 2.598076211 + -1 = 1.598076211 b = 1.598076211 Simplifying b = 1.598076211Subproblem 2
b + 1 = -2.598076211 Simplifying b + 1 = -2.598076211 Reorder the terms: 1 + b = -2.598076211 Solving 1 + b = -2.598076211 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + b = -2.598076211 + -1 Combine like terms: 1 + -1 = 0 0 + b = -2.598076211 + -1 b = -2.598076211 + -1 Combine like terms: -2.598076211 + -1 = -3.598076211 b = -3.598076211 Simplifying b = -3.598076211Solution
The solution to the problem is based on the solutions from the subproblems. b = {1.598076211, -3.598076211}
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