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Simplifying (9b)(9 + -1b) = 32 Remove parenthesis around (9b) 9b(9 + -1b) = 32 (9 * 9b + -1b * 9b) = 32 (81b + -9b2) = 32 Solving 81b + -9b2 = 32 Solving for variable 'b'. Reorder the terms: -32 + 81b + -9b2 = 32 + -32 Combine like terms: 32 + -32 = 0 -32 + 81b + -9b2 = 0 Begin completing the square. Divide all terms by -9 the coefficient of the squared term: Divide each side by '-9'. 3.555555556 + -9b + b2 = 0 Move the constant term to the right: Add '-3.555555556' to each side of the equation. 3.555555556 + -9b + -3.555555556 + b2 = 0 + -3.555555556 Reorder the terms: 3.555555556 + -3.555555556 + -9b + b2 = 0 + -3.555555556 Combine like terms: 3.555555556 + -3.555555556 = 0.000000000 0.000000000 + -9b + b2 = 0 + -3.555555556 -9b + b2 = 0 + -3.555555556 Combine like terms: 0 + -3.555555556 = -3.555555556 -9b + b2 = -3.555555556 The b term is -9b. Take half its coefficient (-4.5). Square it (20.25) and add it to both sides. Add '20.25' to each side of the equation. -9b + 20.25 + b2 = -3.555555556 + 20.25 Reorder the terms: 20.25 + -9b + b2 = -3.555555556 + 20.25 Combine like terms: -3.555555556 + 20.25 = 16.694444444 20.25 + -9b + b2 = 16.694444444 Factor a perfect square on the left side: (b + -4.5)(b + -4.5) = 16.694444444 Calculate the square root of the right side: 4.085883557 Break this problem into two subproblems by setting (b + -4.5) equal to 4.085883557 and -4.085883557.Subproblem 1
b + -4.5 = 4.085883557 Simplifying b + -4.5 = 4.085883557 Reorder the terms: -4.5 + b = 4.085883557 Solving -4.5 + b = 4.085883557 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + b = 4.085883557 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + b = 4.085883557 + 4.5 b = 4.085883557 + 4.5 Combine like terms: 4.085883557 + 4.5 = 8.585883557 b = 8.585883557 Simplifying b = 8.585883557Subproblem 2
b + -4.5 = -4.085883557 Simplifying b + -4.5 = -4.085883557 Reorder the terms: -4.5 + b = -4.085883557 Solving -4.5 + b = -4.085883557 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '4.5' to each side of the equation. -4.5 + 4.5 + b = -4.085883557 + 4.5 Combine like terms: -4.5 + 4.5 = 0.0 0.0 + b = -4.085883557 + 4.5 b = -4.085883557 + 4.5 Combine like terms: -4.085883557 + 4.5 = 0.414116443 b = 0.414116443 Simplifying b = 0.414116443Solution
The solution to the problem is based on the solutions from the subproblems. b = {8.585883557, 0.414116443}
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