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Simplifying 0.5b + 4 = 2b(b + 2) Reorder the terms: 4 + 0.5b = 2b(b + 2) Reorder the terms: 4 + 0.5b = 2b(2 + b) 4 + 0.5b = (2 * 2b + b * 2b) 4 + 0.5b = (4b + 2b2) Solving 4 + 0.5b = 4b + 2b2 Solving for variable 'b'. Combine like terms: 0.5b + -4b = -3.5b 4 + -3.5b + -2b2 = 4b + 2b2 + -4b + -2b2 Reorder the terms: 4 + -3.5b + -2b2 = 4b + -4b + 2b2 + -2b2 Combine like terms: 4b + -4b = 0 4 + -3.5b + -2b2 = 0 + 2b2 + -2b2 4 + -3.5b + -2b2 = 2b2 + -2b2 Combine like terms: 2b2 + -2b2 = 0 4 + -3.5b + -2b2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -2 + 1.75b + b2 = 0 Move the constant term to the right: Add '2' to each side of the equation. -2 + 1.75b + 2 + b2 = 0 + 2 Reorder the terms: -2 + 2 + 1.75b + b2 = 0 + 2 Combine like terms: -2 + 2 = 0 0 + 1.75b + b2 = 0 + 2 1.75b + b2 = 0 + 2 Combine like terms: 0 + 2 = 2 1.75b + b2 = 2 The b term is 1.75b. Take half its coefficient (0.875). Square it (0.765625) and add it to both sides. Add '0.765625' to each side of the equation. 1.75b + 0.765625 + b2 = 2 + 0.765625 Reorder the terms: 0.765625 + 1.75b + b2 = 2 + 0.765625 Combine like terms: 2 + 0.765625 = 2.765625 0.765625 + 1.75b + b2 = 2.765625 Factor a perfect square on the left side: (b + 0.875)(b + 0.875) = 2.765625 Calculate the square root of the right side: 1.663016837 Break this problem into two subproblems by setting (b + 0.875) equal to 1.663016837 and -1.663016837.Subproblem 1
b + 0.875 = 1.663016837 Simplifying b + 0.875 = 1.663016837 Reorder the terms: 0.875 + b = 1.663016837 Solving 0.875 + b = 1.663016837 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-0.875' to each side of the equation. 0.875 + -0.875 + b = 1.663016837 + -0.875 Combine like terms: 0.875 + -0.875 = 0.000 0.000 + b = 1.663016837 + -0.875 b = 1.663016837 + -0.875 Combine like terms: 1.663016837 + -0.875 = 0.788016837 b = 0.788016837 Simplifying b = 0.788016837Subproblem 2
b + 0.875 = -1.663016837 Simplifying b + 0.875 = -1.663016837 Reorder the terms: 0.875 + b = -1.663016837 Solving 0.875 + b = -1.663016837 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-0.875' to each side of the equation. 0.875 + -0.875 + b = -1.663016837 + -0.875 Combine like terms: 0.875 + -0.875 = 0.000 0.000 + b = -1.663016837 + -0.875 b = -1.663016837 + -0.875 Combine like terms: -1.663016837 + -0.875 = -2.538016837 b = -2.538016837 Simplifying b = -2.538016837Solution
The solution to the problem is based on the solutions from the subproblems. b = {0.788016837, -2.538016837}
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