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Simplifying -5b + 24 = -8(b + -6) * 16b Reorder the terms: 24 + -5b = -8(b + -6) * 16b Reorder the terms: 24 + -5b = -8(-6 + b) * 16b Reorder the terms for easier multiplication: 24 + -5b = -8 * 16b(-6 + b) Multiply -8 * 16 24 + -5b = -128b(-6 + b) 24 + -5b = (-6 * -128b + b * -128b) 24 + -5b = (768b + -128b2) Solving 24 + -5b = 768b + -128b2 Solving for variable 'b'. Combine like terms: -5b + -768b = -773b 24 + -773b + 128b2 = 768b + -128b2 + -768b + 128b2 Reorder the terms: 24 + -773b + 128b2 = 768b + -768b + -128b2 + 128b2 Combine like terms: 768b + -768b = 0 24 + -773b + 128b2 = 0 + -128b2 + 128b2 24 + -773b + 128b2 = -128b2 + 128b2 Combine like terms: -128b2 + 128b2 = 0 24 + -773b + 128b2 = 0 Begin completing the square. Divide all terms by 128 the coefficient of the squared term: Divide each side by '128'. 0.1875 + -6.0390625b + b2 = 0 Move the constant term to the right: Add '-0.1875' to each side of the equation. 0.1875 + -6.0390625b + -0.1875 + b2 = 0 + -0.1875 Reorder the terms: 0.1875 + -0.1875 + -6.0390625b + b2 = 0 + -0.1875 Combine like terms: 0.1875 + -0.1875 = 0.0000 0.0000 + -6.0390625b + b2 = 0 + -0.1875 -6.0390625b + b2 = 0 + -0.1875 Combine like terms: 0 + -0.1875 = -0.1875 -6.0390625b + b2 = -0.1875 The b term is -6.0390625b. Take half its coefficient (-3.01953125). Square it (9.117568970) and add it to both sides. Add '9.117568970' to each side of the equation. -6.0390625b + 9.117568970 + b2 = -0.1875 + 9.117568970 Reorder the terms: 9.117568970 + -6.0390625b + b2 = -0.1875 + 9.117568970 Combine like terms: -0.1875 + 9.117568970 = 8.93006897 9.117568970 + -6.0390625b + b2 = 8.93006897 Factor a perfect square on the left side: (b + -3.01953125)(b + -3.01953125) = 8.93006897 Calculate the square root of the right side: 2.988322099 Break this problem into two subproblems by setting (b + -3.01953125) equal to 2.988322099 and -2.988322099.Subproblem 1
b + -3.01953125 = 2.988322099 Simplifying b + -3.01953125 = 2.988322099 Reorder the terms: -3.01953125 + b = 2.988322099 Solving -3.01953125 + b = 2.988322099 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '3.01953125' to each side of the equation. -3.01953125 + 3.01953125 + b = 2.988322099 + 3.01953125 Combine like terms: -3.01953125 + 3.01953125 = 0.00000000 0.00000000 + b = 2.988322099 + 3.01953125 b = 2.988322099 + 3.01953125 Combine like terms: 2.988322099 + 3.01953125 = 6.007853349 b = 6.007853349 Simplifying b = 6.007853349Subproblem 2
b + -3.01953125 = -2.988322099 Simplifying b + -3.01953125 = -2.988322099 Reorder the terms: -3.01953125 + b = -2.988322099 Solving -3.01953125 + b = -2.988322099 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '3.01953125' to each side of the equation. -3.01953125 + 3.01953125 + b = -2.988322099 + 3.01953125 Combine like terms: -3.01953125 + 3.01953125 = 0.00000000 0.00000000 + b = -2.988322099 + 3.01953125 b = -2.988322099 + 3.01953125 Combine like terms: -2.988322099 + 3.01953125 = 0.031209151 b = 0.031209151 Simplifying b = 0.031209151Solution
The solution to the problem is based on the solutions from the subproblems. b = {6.007853349, 0.031209151}
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