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Simplifying 0.25(2b + -12) + b2 = 0.5 + 22 Reorder the terms: 0.25(-12 + 2b) + b2 = 0.5 + 22 (-12 * 0.25 + 2b * 0.25) + b2 = 0.5 + 22 (-3 + 0.5b) + b2 = 0.5 + 22 Combine like terms: 0.5 + 22 = 22.5 -3 + 0.5b + b2 = 22.5 Solving -3 + 0.5b + b2 = 22.5 Solving for variable 'b'. Reorder the terms: -3 + -22.5 + 0.5b + b2 = 22.5 + -22.5 Combine like terms: -3 + -22.5 = -25.5 -25.5 + 0.5b + b2 = 22.5 + -22.5 Combine like terms: 22.5 + -22.5 = 0.0 -25.5 + 0.5b + b2 = 0.0 Begin completing the square. Move the constant term to the right: Add '25.5' to each side of the equation. -25.5 + 0.5b + 25.5 + b2 = 0.0 + 25.5 Reorder the terms: -25.5 + 25.5 + 0.5b + b2 = 0.0 + 25.5 Combine like terms: -25.5 + 25.5 = 0.0 0.0 + 0.5b + b2 = 0.0 + 25.5 0.5b + b2 = 0.0 + 25.5 Combine like terms: 0.0 + 25.5 = 25.5 0.5b + b2 = 25.5 The b term is 0.5b. Take half its coefficient (0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. 0.5b + 0.0625 + b2 = 25.5 + 0.0625 Reorder the terms: 0.0625 + 0.5b + b2 = 25.5 + 0.0625 Combine like terms: 25.5 + 0.0625 = 25.5625 0.0625 + 0.5b + b2 = 25.5625 Factor a perfect square on the left side: (b + 0.25)(b + 0.25) = 25.5625 Calculate the square root of the right side: 5.055937104 Break this problem into two subproblems by setting (b + 0.25) equal to 5.055937104 and -5.055937104.Subproblem 1
b + 0.25 = 5.055937104 Simplifying b + 0.25 = 5.055937104 Reorder the terms: 0.25 + b = 5.055937104 Solving 0.25 + b = 5.055937104 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + b = 5.055937104 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + b = 5.055937104 + -0.25 b = 5.055937104 + -0.25 Combine like terms: 5.055937104 + -0.25 = 4.805937104 b = 4.805937104 Simplifying b = 4.805937104Subproblem 2
b + 0.25 = -5.055937104 Simplifying b + 0.25 = -5.055937104 Reorder the terms: 0.25 + b = -5.055937104 Solving 0.25 + b = -5.055937104 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + b = -5.055937104 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + b = -5.055937104 + -0.25 b = -5.055937104 + -0.25 Combine like terms: -5.055937104 + -0.25 = -5.305937104 b = -5.305937104 Simplifying b = -5.305937104Solution
The solution to the problem is based on the solutions from the subproblems. b = {4.805937104, -5.305937104}
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