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Simplifying b(b + 3) + (2b + -5) = 38 Reorder the terms: b(3 + b) + (2b + -5) = 38 (3 * b + b * b) + (2b + -5) = 38 (3b + b2) + (2b + -5) = 38 Reorder the terms: 3b + b2 + (-5 + 2b) = 38 Remove parenthesis around (-5 + 2b) 3b + b2 + -5 + 2b = 38 Reorder the terms: -5 + 3b + 2b + b2 = 38 Combine like terms: 3b + 2b = 5b -5 + 5b + b2 = 38 Solving -5 + 5b + b2 = 38 Solving for variable 'b'. Reorder the terms: -5 + -38 + 5b + b2 = 38 + -38 Combine like terms: -5 + -38 = -43 -43 + 5b + b2 = 38 + -38 Combine like terms: 38 + -38 = 0 -43 + 5b + b2 = 0 Begin completing the square. Move the constant term to the right: Add '43' to each side of the equation. -43 + 5b + 43 + b2 = 0 + 43 Reorder the terms: -43 + 43 + 5b + b2 = 0 + 43 Combine like terms: -43 + 43 = 0 0 + 5b + b2 = 0 + 43 5b + b2 = 0 + 43 Combine like terms: 0 + 43 = 43 5b + b2 = 43 The b term is 5b. Take half its coefficient (2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. 5b + 6.25 + b2 = 43 + 6.25 Reorder the terms: 6.25 + 5b + b2 = 43 + 6.25 Combine like terms: 43 + 6.25 = 49.25 6.25 + 5b + b2 = 49.25 Factor a perfect square on the left side: (b + 2.5)(b + 2.5) = 49.25 Calculate the square root of the right side: 7.017834424 Break this problem into two subproblems by setting (b + 2.5) equal to 7.017834424 and -7.017834424.Subproblem 1
b + 2.5 = 7.017834424 Simplifying b + 2.5 = 7.017834424 Reorder the terms: 2.5 + b = 7.017834424 Solving 2.5 + b = 7.017834424 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + b = 7.017834424 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + b = 7.017834424 + -2.5 b = 7.017834424 + -2.5 Combine like terms: 7.017834424 + -2.5 = 4.517834424 b = 4.517834424 Simplifying b = 4.517834424Subproblem 2
b + 2.5 = -7.017834424 Simplifying b + 2.5 = -7.017834424 Reorder the terms: 2.5 + b = -7.017834424 Solving 2.5 + b = -7.017834424 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-2.5' to each side of the equation. 2.5 + -2.5 + b = -7.017834424 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + b = -7.017834424 + -2.5 b = -7.017834424 + -2.5 Combine like terms: -7.017834424 + -2.5 = -9.517834424 b = -9.517834424 Simplifying b = -9.517834424Solution
The solution to the problem is based on the solutions from the subproblems. b = {4.517834424, -9.517834424}
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