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Simplifying 2b(5 + -1b) = 9 (5 * 2b + -1b * 2b) = 9 (10b + -2b2) = 9 Solving 10b + -2b2 = 9 Solving for variable 'b'. Reorder the terms: -9 + 10b + -2b2 = 9 + -9 Combine like terms: 9 + -9 = 0 -9 + 10b + -2b2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. 4.5 + -5b + b2 = 0 Move the constant term to the right: Add '-4.5' to each side of the equation. 4.5 + -5b + -4.5 + b2 = 0 + -4.5 Reorder the terms: 4.5 + -4.5 + -5b + b2 = 0 + -4.5 Combine like terms: 4.5 + -4.5 = 0.0 0.0 + -5b + b2 = 0 + -4.5 -5b + b2 = 0 + -4.5 Combine like terms: 0 + -4.5 = -4.5 -5b + b2 = -4.5 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 = -4.5 + 6.25 Reorder the terms: 6.25 + -5b + b2 = -4.5 + 6.25 Combine like terms: -4.5 + 6.25 = 1.75 6.25 + -5b + b2 = 1.75 Factor a perfect square on the left side: (b + -2.5)(b + -2.5) = 1.75 Calculate the square root of the right side: 1.322875656 Break this problem into two subproblems by setting (b + -2.5) equal to 1.322875656 and -1.322875656.Subproblem 1
b + -2.5 = 1.322875656 Simplifying b + -2.5 = 1.322875656 Reorder the terms: -2.5 + b = 1.322875656 Solving -2.5 + b = 1.322875656 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 = 1.322875656 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + b = 1.322875656 + 2.5 b = 1.322875656 + 2.5 Combine like terms: 1.322875656 + 2.5 = 3.822875656 b = 3.822875656 Simplifying b = 3.822875656Subproblem 2
b + -2.5 = -1.322875656 Simplifying b + -2.5 = -1.322875656 Reorder the terms: -2.5 + b = -1.322875656 Solving -2.5 + b = -1.322875656 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 = -1.322875656 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + b = -1.322875656 + 2.5 b = -1.322875656 + 2.5 Combine like terms: -1.322875656 + 2.5 = 1.177124344 b = 1.177124344 Simplifying b = 1.177124344Solution
The solution to the problem is based on the solutions from the subproblems. b = {3.822875656, 1.177124344}
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