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Simplifying 0 = 0.5b2 + 2.5b + -324 Reorder the terms: 0 = -324 + 2.5b + 0.5b2 Solving 0 = -324 + 2.5b + 0.5b2 Solving for variable 'b'. Combine like terms: 0 + 324 = 324 324 + -2.5b + -0.5b2 = -324 + 2.5b + 0.5b2 + 324 + -2.5b + -0.5b2 Reorder the terms: 324 + -2.5b + -0.5b2 = -324 + 324 + 2.5b + -2.5b + 0.5b2 + -0.5b2 Combine like terms: -324 + 324 = 0 324 + -2.5b + -0.5b2 = 0 + 2.5b + -2.5b + 0.5b2 + -0.5b2 324 + -2.5b + -0.5b2 = 2.5b + -2.5b + 0.5b2 + -0.5b2 Combine like terms: 2.5b + -2.5b = 0.0 324 + -2.5b + -0.5b2 = 0.0 + 0.5b2 + -0.5b2 324 + -2.5b + -0.5b2 = 0.5b2 + -0.5b2 Combine like terms: 0.5b2 + -0.5b2 = 0.0 324 + -2.5b + -0.5b2 = 0.0 Begin completing the square. Divide all terms by -0.5 the coefficient of the squared term: Divide each side by '-0.5'. -648 + 5b + b2 = 0 Move the constant term to the right: Add '648' to each side of the equation. -648 + 5b + 648 + b2 = 0 + 648 Reorder the terms: -648 + 648 + 5b + b2 = 0 + 648 Combine like terms: -648 + 648 = 0 0 + 5b + b2 = 0 + 648 5b + b2 = 0 + 648 Combine like terms: 0 + 648 = 648 5b + b2 = 648 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 = 648 + 6.25 Reorder the terms: 6.25 + 5b + b2 = 648 + 6.25 Combine like terms: 648 + 6.25 = 654.25 6.25 + 5b + b2 = 654.25 Factor a perfect square on the left side: (b + 2.5)(b + 2.5) = 654.25 Calculate the square root of the right side: 25.578311125 Break this problem into two subproblems by setting (b + 2.5) equal to 25.578311125 and -25.578311125.Subproblem 1
b + 2.5 = 25.578311125 Simplifying b + 2.5 = 25.578311125 Reorder the terms: 2.5 + b = 25.578311125 Solving 2.5 + b = 25.578311125 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 = 25.578311125 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + b = 25.578311125 + -2.5 b = 25.578311125 + -2.5 Combine like terms: 25.578311125 + -2.5 = 23.078311125 b = 23.078311125 Simplifying b = 23.078311125Subproblem 2
b + 2.5 = -25.578311125 Simplifying b + 2.5 = -25.578311125 Reorder the terms: 2.5 + b = -25.578311125 Solving 2.5 + b = -25.578311125 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 = -25.578311125 + -2.5 Combine like terms: 2.5 + -2.5 = 0.0 0.0 + b = -25.578311125 + -2.5 b = -25.578311125 + -2.5 Combine like terms: -25.578311125 + -2.5 = -28.078311125 b = -28.078311125 Simplifying b = -28.078311125Solution
The solution to the problem is based on the solutions from the subproblems. b = {23.078311125, -28.078311125}
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