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Simplifying 52b2 + -156b + -407 = 0 Reorder the terms: -407 + -156b + 52b2 = 0 Solving -407 + -156b + 52b2 = 0 Solving for variable 'b'. Begin completing the square. Divide all terms by 52 the coefficient of the squared term: Divide each side by '52'. -7.826923077 + -3b + b2 = 0 Move the constant term to the right: Add '7.826923077' to each side of the equation. -7.826923077 + -3b + 7.826923077 + b2 = 0 + 7.826923077 Reorder the terms: -7.826923077 + 7.826923077 + -3b + b2 = 0 + 7.826923077 Combine like terms: -7.826923077 + 7.826923077 = 0.000000000 0.000000000 + -3b + b2 = 0 + 7.826923077 -3b + b2 = 0 + 7.826923077 Combine like terms: 0 + 7.826923077 = 7.826923077 -3b + b2 = 7.826923077 The b term is -3b. Take half its coefficient (-1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. -3b + 2.25 + b2 = 7.826923077 + 2.25 Reorder the terms: 2.25 + -3b + b2 = 7.826923077 + 2.25 Combine like terms: 7.826923077 + 2.25 = 10.076923077 2.25 + -3b + b2 = 10.076923077 Factor a perfect square on the left side: (b + -1.5)(b + -1.5) = 10.076923077 Calculate the square root of the right side: 3.174416966 Break this problem into two subproblems by setting (b + -1.5) equal to 3.174416966 and -3.174416966.Subproblem 1
b + -1.5 = 3.174416966 Simplifying b + -1.5 = 3.174416966 Reorder the terms: -1.5 + b = 3.174416966 Solving -1.5 + b = 3.174416966 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + b = 3.174416966 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + b = 3.174416966 + 1.5 b = 3.174416966 + 1.5 Combine like terms: 3.174416966 + 1.5 = 4.674416966 b = 4.674416966 Simplifying b = 4.674416966Subproblem 2
b + -1.5 = -3.174416966 Simplifying b + -1.5 = -3.174416966 Reorder the terms: -1.5 + b = -3.174416966 Solving -1.5 + b = -3.174416966 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + b = -3.174416966 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + b = -3.174416966 + 1.5 b = -3.174416966 + 1.5 Combine like terms: -3.174416966 + 1.5 = -1.674416966 b = -1.674416966 Simplifying b = -1.674416966Solution
The solution to the problem is based on the solutions from the subproblems. b = {4.674416966, -1.674416966}
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