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Simplifying 9b2 + -7b + -24 = -11b Reorder the terms: -24 + -7b + 9b2 = -11b Solving -24 + -7b + 9b2 = -11b Solving for variable 'b'. Reorder the terms: -24 + -7b + 11b + 9b2 = -11b + 11b Combine like terms: -7b + 11b = 4b -24 + 4b + 9b2 = -11b + 11b Combine like terms: -11b + 11b = 0 -24 + 4b + 9b2 = 0 Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. -2.666666667 + 0.4444444444b + b2 = 0 Move the constant term to the right: Add '2.666666667' to each side of the equation. -2.666666667 + 0.4444444444b + 2.666666667 + b2 = 0 + 2.666666667 Reorder the terms: -2.666666667 + 2.666666667 + 0.4444444444b + b2 = 0 + 2.666666667 Combine like terms: -2.666666667 + 2.666666667 = 0.000000000 0.000000000 + 0.4444444444b + b2 = 0 + 2.666666667 0.4444444444b + b2 = 0 + 2.666666667 Combine like terms: 0 + 2.666666667 = 2.666666667 0.4444444444b + b2 = 2.666666667 The b term is 0.4444444444b. Take half its coefficient (0.2222222222). Square it (0.04938271604) and add it to both sides. Add '0.04938271604' to each side of the equation. 0.4444444444b + 0.04938271604 + b2 = 2.666666667 + 0.04938271604 Reorder the terms: 0.04938271604 + 0.4444444444b + b2 = 2.666666667 + 0.04938271604 Combine like terms: 2.666666667 + 0.04938271604 = 2.71604938304 0.04938271604 + 0.4444444444b + b2 = 2.71604938304 Factor a perfect square on the left side: (b + 0.2222222222)(b + 0.2222222222) = 2.71604938304 Calculate the square root of the right side: 1.648044108 Break this problem into two subproblems by setting (b + 0.2222222222) equal to 1.648044108 and -1.648044108.Subproblem 1
b + 0.2222222222 = 1.648044108 Simplifying b + 0.2222222222 = 1.648044108 Reorder the terms: 0.2222222222 + b = 1.648044108 Solving 0.2222222222 + b = 1.648044108 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-0.2222222222' to each side of the equation. 0.2222222222 + -0.2222222222 + b = 1.648044108 + -0.2222222222 Combine like terms: 0.2222222222 + -0.2222222222 = 0.0000000000 0.0000000000 + b = 1.648044108 + -0.2222222222 b = 1.648044108 + -0.2222222222 Combine like terms: 1.648044108 + -0.2222222222 = 1.4258218858 b = 1.4258218858 Simplifying b = 1.4258218858Subproblem 2
b + 0.2222222222 = -1.648044108 Simplifying b + 0.2222222222 = -1.648044108 Reorder the terms: 0.2222222222 + b = -1.648044108 Solving 0.2222222222 + b = -1.648044108 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-0.2222222222' to each side of the equation. 0.2222222222 + -0.2222222222 + b = -1.648044108 + -0.2222222222 Combine like terms: 0.2222222222 + -0.2222222222 = 0.0000000000 0.0000000000 + b = -1.648044108 + -0.2222222222 b = -1.648044108 + -0.2222222222 Combine like terms: -1.648044108 + -0.2222222222 = -1.8702663302 b = -1.8702663302 Simplifying b = -1.8702663302Solution
The solution to the problem is based on the solutions from the subproblems. b = {1.4258218858, -1.8702663302}
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