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Simplifying 0.125(2b + -12) + b2 = 0.500b + 22 Reorder the terms: 0.125(-12 + 2b) + b2 = 0.500b + 22 (-12 * 0.125 + 2b * 0.125) + b2 = 0.500b + 22 (-1.5 + 0.25b) + b2 = 0.500b + 22 Reorder the terms: -1.5 + 0.25b + b2 = 22 + 0.500b Solving -1.5 + 0.25b + b2 = 22 + 0.500b Solving for variable 'b'. Reorder the terms: -1.5 + -22 + 0.25b + -0.500b + b2 = 22 + 0.500b + -22 + -0.500b Combine like terms: -1.5 + -22 = -23.5 -23.5 + 0.25b + -0.500b + b2 = 22 + 0.500b + -22 + -0.500b Combine like terms: 0.25b + -0.500b = -0.25b -23.5 + -0.25b + b2 = 22 + 0.500b + -22 + -0.500b Reorder the terms: -23.5 + -0.25b + b2 = 22 + -22 + 0.500b + -0.500b Combine like terms: 22 + -22 = 0 -23.5 + -0.25b + b2 = 0 + 0.500b + -0.500b -23.5 + -0.25b + b2 = 0.500b + -0.500b Combine like terms: 0.500b + -0.500b = 0.000 -23.5 + -0.25b + b2 = 0.000 Begin completing the square. Move the constant term to the right: Add '23.5' to each side of the equation. -23.5 + -0.25b + 23.5 + b2 = 0.000 + 23.5 Reorder the terms: -23.5 + 23.5 + -0.25b + b2 = 0.000 + 23.5 Combine like terms: -23.5 + 23.5 = 0.0 0.0 + -0.25b + b2 = 0.000 + 23.5 -0.25b + b2 = 0.000 + 23.5 Combine like terms: 0.000 + 23.5 = 23.5 -0.25b + b2 = 23.5 The b term is -0.25b. Take half its coefficient (-0.125). Square it (0.015625) and add it to both sides. Add '0.015625' to each side of the equation. -0.25b + 0.015625 + b2 = 23.5 + 0.015625 Reorder the terms: 0.015625 + -0.25b + b2 = 23.5 + 0.015625 Combine like terms: 23.5 + 0.015625 = 23.515625 0.015625 + -0.25b + b2 = 23.515625 Factor a perfect square on the left side: (b + -0.125)(b + -0.125) = 23.515625 Calculate the square root of the right side: 4.849291185 Break this problem into two subproblems by setting (b + -0.125) equal to 4.849291185 and -4.849291185.Subproblem 1
b + -0.125 = 4.849291185 Simplifying b + -0.125 = 4.849291185 Reorder the terms: -0.125 + b = 4.849291185 Solving -0.125 + b = 4.849291185 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '0.125' to each side of the equation. -0.125 + 0.125 + b = 4.849291185 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + b = 4.849291185 + 0.125 b = 4.849291185 + 0.125 Combine like terms: 4.849291185 + 0.125 = 4.974291185 b = 4.974291185 Simplifying b = 4.974291185Subproblem 2
b + -0.125 = -4.849291185 Simplifying b + -0.125 = -4.849291185 Reorder the terms: -0.125 + b = -4.849291185 Solving -0.125 + b = -4.849291185 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '0.125' to each side of the equation. -0.125 + 0.125 + b = -4.849291185 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + b = -4.849291185 + 0.125 b = -4.849291185 + 0.125 Combine like terms: -4.849291185 + 0.125 = -4.724291185 b = -4.724291185 Simplifying b = -4.724291185Solution
The solution to the problem is based on the solutions from the subproblems. b = {4.974291185, -4.724291185}
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