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Simplifying 9 = 8b(2b + -3) Reorder the terms: 9 = 8b(-3 + 2b) 9 = (-3 * 8b + 2b * 8b) 9 = (-24b + 16b2) Solving 9 = -24b + 16b2 Solving for variable 'b'. Reorder the terms: 9 + 24b + -16b2 = -24b + 24b + 16b2 + -16b2 Combine like terms: -24b + 24b = 0 9 + 24b + -16b2 = 0 + 16b2 + -16b2 9 + 24b + -16b2 = 16b2 + -16b2 Combine like terms: 16b2 + -16b2 = 0 9 + 24b + -16b2 = 0 Begin completing the square. Divide all terms by -16 the coefficient of the squared term: Divide each side by '-16'. -0.5625 + -1.5b + b2 = 0 Move the constant term to the right: Add '0.5625' to each side of the equation. -0.5625 + -1.5b + 0.5625 + b2 = 0 + 0.5625 Reorder the terms: -0.5625 + 0.5625 + -1.5b + b2 = 0 + 0.5625 Combine like terms: -0.5625 + 0.5625 = 0.0000 0.0000 + -1.5b + b2 = 0 + 0.5625 -1.5b + b2 = 0 + 0.5625 Combine like terms: 0 + 0.5625 = 0.5625 -1.5b + b2 = 0.5625 The b term is -1.5b. Take half its coefficient (-0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. -1.5b + 0.5625 + b2 = 0.5625 + 0.5625 Reorder the terms: 0.5625 + -1.5b + b2 = 0.5625 + 0.5625 Combine like terms: 0.5625 + 0.5625 = 1.125 0.5625 + -1.5b + b2 = 1.125 Factor a perfect square on the left side: (b + -0.75)(b + -0.75) = 1.125 Calculate the square root of the right side: 1.060660172 Break this problem into two subproblems by setting (b + -0.75) equal to 1.060660172 and -1.060660172.Subproblem 1
b + -0.75 = 1.060660172 Simplifying b + -0.75 = 1.060660172 Reorder the terms: -0.75 + b = 1.060660172 Solving -0.75 + b = 1.060660172 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + b = 1.060660172 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + b = 1.060660172 + 0.75 b = 1.060660172 + 0.75 Combine like terms: 1.060660172 + 0.75 = 1.810660172 b = 1.810660172 Simplifying b = 1.810660172Subproblem 2
b + -0.75 = -1.060660172 Simplifying b + -0.75 = -1.060660172 Reorder the terms: -0.75 + b = -1.060660172 Solving -0.75 + b = -1.060660172 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '0.75' to each side of the equation. -0.75 + 0.75 + b = -1.060660172 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + b = -1.060660172 + 0.75 b = -1.060660172 + 0.75 Combine like terms: -1.060660172 + 0.75 = -0.310660172 b = -0.310660172 Simplifying b = -0.310660172Solution
The solution to the problem is based on the solutions from the subproblems. b = {1.810660172, -0.310660172}
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