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Simplifying 9b2 + 9b = 24 Reorder the terms: 9b + 9b2 = 24 Solving 9b + 9b2 = 24 Solving for variable 'b'. Reorder the terms: -24 + 9b + 9b2 = 24 + -24 Combine like terms: 24 + -24 = 0 -24 + 9b + 9b2 = 0 Factor out the Greatest Common Factor (GCF), '3'. 3(-8 + 3b + 3b2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(-8 + 3b + 3b2)' equal to zero and attempt to solve: Simplifying -8 + 3b + 3b2 = 0 Solving -8 + 3b + 3b2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -2.666666667 + b + b2 = 0 Move the constant term to the right: Add '2.666666667' to each side of the equation. -2.666666667 + b + 2.666666667 + b2 = 0 + 2.666666667 Reorder the terms: -2.666666667 + 2.666666667 + b + b2 = 0 + 2.666666667 Combine like terms: -2.666666667 + 2.666666667 = 0.000000000 0.000000000 + b + b2 = 0 + 2.666666667 b + b2 = 0 + 2.666666667 Combine like terms: 0 + 2.666666667 = 2.666666667 b + b2 = 2.666666667 The b term is b. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. + 0.25 + b2 = 2.666666667 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + b2 = 2.666666667 + 0.25 Combine like terms: 2.666666667 + 0.25 = 2.916666667 1.25 + b2 = 2.916666667 Factor a perfect square on the left side: (b + 0.5)(b + 0.5) = 2.916666667 Calculate the square root of the right side: 1.707825128 Break this problem into two subproblems by setting (b + 0.5) equal to 1.707825128 and -1.707825128.Subproblem 1
b + 0.5 = 1.707825128 Simplifying b + 0.5 = 1.707825128 Reorder the terms: 0.5 + b = 1.707825128 Solving 0.5 + b = 1.707825128 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + b = 1.707825128 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + b = 1.707825128 + -0.5 b = 1.707825128 + -0.5 Combine like terms: 1.707825128 + -0.5 = 1.207825128 b = 1.207825128 Simplifying b = 1.207825128Subproblem 2
b + 0.5 = -1.707825128 Simplifying b + 0.5 = -1.707825128 Reorder the terms: 0.5 + b = -1.707825128 Solving 0.5 + b = -1.707825128 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + b = -1.707825128 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + b = -1.707825128 + -0.5 b = -1.707825128 + -0.5 Combine like terms: -1.707825128 + -0.5 = -2.207825128 b = -2.207825128 Simplifying b = -2.207825128Solution
The solution to the problem is based on the solutions from the subproblems. b = {1.207825128, -2.207825128}Solution
b = {1.207825128, -2.207825128}
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