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Simplifying 16b2 + 162b + 180 = 0 Reorder the terms: 180 + 162b + 16b2 = 0 Solving 180 + 162b + 16b2 = 0 Solving for variable 'b'. Factor out the Greatest Common Factor (GCF), '2'. 2(90 + 81b + 8b2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(90 + 81b + 8b2)' equal to zero and attempt to solve: Simplifying 90 + 81b + 8b2 = 0 Solving 90 + 81b + 8b2 = 0 Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. 11.25 + 10.125b + b2 = 0 Move the constant term to the right: Add '-11.25' to each side of the equation. 11.25 + 10.125b + -11.25 + b2 = 0 + -11.25 Reorder the terms: 11.25 + -11.25 + 10.125b + b2 = 0 + -11.25 Combine like terms: 11.25 + -11.25 = 0.00 0.00 + 10.125b + b2 = 0 + -11.25 10.125b + b2 = 0 + -11.25 Combine like terms: 0 + -11.25 = -11.25 10.125b + b2 = -11.25 The b term is 10.125b. Take half its coefficient (5.0625). Square it (25.62890625) and add it to both sides. Add '25.62890625' to each side of the equation. 10.125b + 25.62890625 + b2 = -11.25 + 25.62890625 Reorder the terms: 25.62890625 + 10.125b + b2 = -11.25 + 25.62890625 Combine like terms: -11.25 + 25.62890625 = 14.37890625 25.62890625 + 10.125b + b2 = 14.37890625 Factor a perfect square on the left side: (b + 5.0625)(b + 5.0625) = 14.37890625 Calculate the square root of the right side: 3.791952828 Break this problem into two subproblems by setting (b + 5.0625) equal to 3.791952828 and -3.791952828.Subproblem 1
b + 5.0625 = 3.791952828 Simplifying b + 5.0625 = 3.791952828 Reorder the terms: 5.0625 + b = 3.791952828 Solving 5.0625 + b = 3.791952828 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-5.0625' to each side of the equation. 5.0625 + -5.0625 + b = 3.791952828 + -5.0625 Combine like terms: 5.0625 + -5.0625 = 0.0000 0.0000 + b = 3.791952828 + -5.0625 b = 3.791952828 + -5.0625 Combine like terms: 3.791952828 + -5.0625 = -1.270547172 b = -1.270547172 Simplifying b = -1.270547172Subproblem 2
b + 5.0625 = -3.791952828 Simplifying b + 5.0625 = -3.791952828 Reorder the terms: 5.0625 + b = -3.791952828 Solving 5.0625 + b = -3.791952828 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-5.0625' to each side of the equation. 5.0625 + -5.0625 + b = -3.791952828 + -5.0625 Combine like terms: 5.0625 + -5.0625 = 0.0000 0.0000 + b = -3.791952828 + -5.0625 b = -3.791952828 + -5.0625 Combine like terms: -3.791952828 + -5.0625 = -8.854452828 b = -8.854452828 Simplifying b = -8.854452828Solution
The solution to the problem is based on the solutions from the subproblems. b = {-1.270547172, -8.854452828}Solution
b = {-1.270547172, -8.854452828}
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