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Simplifying 4b2 + -24 = b2 + 37b Reorder the terms: -24 + 4b2 = b2 + 37b Reorder the terms: -24 + 4b2 = 37b + b2 Solving -24 + 4b2 = 37b + b2 Solving for variable 'b'. Reorder the terms: -24 + -37b + 4b2 + -1b2 = 37b + b2 + -37b + -1b2 Combine like terms: 4b2 + -1b2 = 3b2 -24 + -37b + 3b2 = 37b + b2 + -37b + -1b2 Reorder the terms: -24 + -37b + 3b2 = 37b + -37b + b2 + -1b2 Combine like terms: 37b + -37b = 0 -24 + -37b + 3b2 = 0 + b2 + -1b2 -24 + -37b + 3b2 = b2 + -1b2 Combine like terms: b2 + -1b2 = 0 -24 + -37b + 3b2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -8 + -12.33333333b + b2 = 0 Move the constant term to the right: Add '8' to each side of the equation. -8 + -12.33333333b + 8 + b2 = 0 + 8 Reorder the terms: -8 + 8 + -12.33333333b + b2 = 0 + 8 Combine like terms: -8 + 8 = 0 0 + -12.33333333b + b2 = 0 + 8 -12.33333333b + b2 = 0 + 8 Combine like terms: 0 + 8 = 8 -12.33333333b + b2 = 8 The b term is -12.33333333b. Take half its coefficient (-6.166666665). Square it (38.02777776) and add it to both sides. Add '38.02777776' to each side of the equation. -12.33333333b + 38.02777776 + b2 = 8 + 38.02777776 Reorder the terms: 38.02777776 + -12.33333333b + b2 = 8 + 38.02777776 Combine like terms: 8 + 38.02777776 = 46.02777776 38.02777776 + -12.33333333b + b2 = 46.02777776 Factor a perfect square on the left side: (b + -6.166666665)(b + -6.166666665) = 46.02777776 Calculate the square root of the right side: 6.784377478 Break this problem into two subproblems by setting (b + -6.166666665) equal to 6.784377478 and -6.784377478.Subproblem 1
b + -6.166666665 = 6.784377478 Simplifying b + -6.166666665 = 6.784377478 Reorder the terms: -6.166666665 + b = 6.784377478 Solving -6.166666665 + b = 6.784377478 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '6.166666665' to each side of the equation. -6.166666665 + 6.166666665 + b = 6.784377478 + 6.166666665 Combine like terms: -6.166666665 + 6.166666665 = 0.000000000 0.000000000 + b = 6.784377478 + 6.166666665 b = 6.784377478 + 6.166666665 Combine like terms: 6.784377478 + 6.166666665 = 12.951044143 b = 12.951044143 Simplifying b = 12.951044143Subproblem 2
b + -6.166666665 = -6.784377478 Simplifying b + -6.166666665 = -6.784377478 Reorder the terms: -6.166666665 + b = -6.784377478 Solving -6.166666665 + b = -6.784377478 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '6.166666665' to each side of the equation. -6.166666665 + 6.166666665 + b = -6.784377478 + 6.166666665 Combine like terms: -6.166666665 + 6.166666665 = 0.000000000 0.000000000 + b = -6.784377478 + 6.166666665 b = -6.784377478 + 6.166666665 Combine like terms: -6.784377478 + 6.166666665 = -0.617710813 b = -0.617710813 Simplifying b = -0.617710813Solution
The solution to the problem is based on the solutions from the subproblems. b = {12.951044143, -0.617710813}
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