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Simplifying 0.5(2b + -12) + b2 = 0.5 + 22 Reorder the terms: 0.5(-12 + 2b) + b2 = 0.5 + 22 (-12 * 0.5 + 2b * 0.5) + b2 = 0.5 + 22 (-6 + 1b) + b2 = 0.5 + 22 Combine like terms: 0.5 + 22 = 22.5 -6 + 1b + b2 = 22.5 Solving -6 + 1b + b2 = 22.5 Solving for variable 'b'. Reorder the terms: -6 + -22.5 + 1b + b2 = 22.5 + -22.5 Combine like terms: -6 + -22.5 = -28.5 -28.5 + 1b + b2 = 22.5 + -22.5 Combine like terms: 22.5 + -22.5 = 0.0 -28.5 + 1b + b2 = 0.0 Begin completing the square. Move the constant term to the right: Add '28.5' to each side of the equation. -28.5 + 1b + 28.5 + b2 = 0.0 + 28.5 Reorder the terms: -28.5 + 28.5 + 1b + b2 = 0.0 + 28.5 Combine like terms: -28.5 + 28.5 = 0.0 0.0 + 1b + b2 = 0.0 + 28.5 1b + b2 = 0.0 + 28.5 Combine like terms: 0.0 + 28.5 = 28.5 1b + b2 = 28.5 The b term is 1b. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. b + 0.25 + b2 = 28.5 + 0.25 Reorder the terms: 0.25 + b + b2 = 28.5 + 0.25 Combine like terms: 28.5 + 0.25 = 28.75 0.25 + b + b2 = 28.75 Factor a perfect square on the left side: (b + 0.5)(b + 0.5) = 28.75 Calculate the square root of the right side: 5.361902647 Break this problem into two subproblems by setting (b + 0.5) equal to 5.361902647 and -5.361902647.Subproblem 1
b + 0.5 = 5.361902647 Simplifying b + 0.5 = 5.361902647 Reorder the terms: 0.5 + b = 5.361902647 Solving 0.5 + b = 5.361902647 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + b = 5.361902647 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + b = 5.361902647 + -0.5 b = 5.361902647 + -0.5 Combine like terms: 5.361902647 + -0.5 = 4.861902647 b = 4.861902647 Simplifying b = 4.861902647Subproblem 2
b + 0.5 = -5.361902647 Simplifying b + 0.5 = -5.361902647 Reorder the terms: 0.5 + b = -5.361902647 Solving 0.5 + b = -5.361902647 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + b = -5.361902647 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + b = -5.361902647 + -0.5 b = -5.361902647 + -0.5 Combine like terms: -5.361902647 + -0.5 = -5.861902647 b = -5.861902647 Simplifying b = -5.861902647Solution
The solution to the problem is based on the solutions from the subproblems. b = {4.861902647, -5.861902647}
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