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Simplifying 3b(b + 4) = 24 + -2(b + -3) Reorder the terms: 3b(4 + b) = 24 + -2(b + -3) (4 * 3b + b * 3b) = 24 + -2(b + -3) (12b + 3b2) = 24 + -2(b + -3) Reorder the terms: 12b + 3b2 = 24 + -2(-3 + b) 12b + 3b2 = 24 + (-3 * -2 + b * -2) 12b + 3b2 = 24 + (6 + -2b) Combine like terms: 24 + 6 = 30 12b + 3b2 = 30 + -2b Solving 12b + 3b2 = 30 + -2b Solving for variable 'b'. Reorder the terms: -30 + 12b + 2b + 3b2 = 30 + -2b + -30 + 2b Combine like terms: 12b + 2b = 14b -30 + 14b + 3b2 = 30 + -2b + -30 + 2b Reorder the terms: -30 + 14b + 3b2 = 30 + -30 + -2b + 2b Combine like terms: 30 + -30 = 0 -30 + 14b + 3b2 = 0 + -2b + 2b -30 + 14b + 3b2 = -2b + 2b Combine like terms: -2b + 2b = 0 -30 + 14b + 3b2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -10 + 4.666666667b + b2 = 0 Move the constant term to the right: Add '10' to each side of the equation. -10 + 4.666666667b + 10 + b2 = 0 + 10 Reorder the terms: -10 + 10 + 4.666666667b + b2 = 0 + 10 Combine like terms: -10 + 10 = 0 0 + 4.666666667b + b2 = 0 + 10 4.666666667b + b2 = 0 + 10 Combine like terms: 0 + 10 = 10 4.666666667b + b2 = 10 The b term is 4.666666667b. Take half its coefficient (2.333333334). Square it (5.444444448) and add it to both sides. Add '5.444444448' to each side of the equation. 4.666666667b + 5.444444448 + b2 = 10 + 5.444444448 Reorder the terms: 5.444444448 + 4.666666667b + b2 = 10 + 5.444444448 Combine like terms: 10 + 5.444444448 = 15.444444448 5.444444448 + 4.666666667b + b2 = 15.444444448 Factor a perfect square on the left side: (b + 2.333333334)(b + 2.333333334) = 15.444444448 Calculate the square root of the right side: 3.929942041 Break this problem into two subproblems by setting (b + 2.333333334) equal to 3.929942041 and -3.929942041.Subproblem 1
b + 2.333333334 = 3.929942041 Simplifying b + 2.333333334 = 3.929942041 Reorder the terms: 2.333333334 + b = 3.929942041 Solving 2.333333334 + b = 3.929942041 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-2.333333334' to each side of the equation. 2.333333334 + -2.333333334 + b = 3.929942041 + -2.333333334 Combine like terms: 2.333333334 + -2.333333334 = 0.000000000 0.000000000 + b = 3.929942041 + -2.333333334 b = 3.929942041 + -2.333333334 Combine like terms: 3.929942041 + -2.333333334 = 1.596608707 b = 1.596608707 Simplifying b = 1.596608707Subproblem 2
b + 2.333333334 = -3.929942041 Simplifying b + 2.333333334 = -3.929942041 Reorder the terms: 2.333333334 + b = -3.929942041 Solving 2.333333334 + b = -3.929942041 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-2.333333334' to each side of the equation. 2.333333334 + -2.333333334 + b = -3.929942041 + -2.333333334 Combine like terms: 2.333333334 + -2.333333334 = 0.000000000 0.000000000 + b = -3.929942041 + -2.333333334 b = -3.929942041 + -2.333333334 Combine like terms: -3.929942041 + -2.333333334 = -6.263275375 b = -6.263275375 Simplifying b = -6.263275375Solution
The solution to the problem is based on the solutions from the subproblems. b = {1.596608707, -6.263275375}
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