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Simplifying -36b2 + 2b + 54 = 0 Reorder the terms: 54 + 2b + -36b2 = 0 Solving 54 + 2b + -36b2 = 0 Solving for variable 'b'. Factor out the Greatest Common Factor (GCF), '2'. 2(27 + b + -18b2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(27 + b + -18b2)' equal to zero and attempt to solve: Simplifying 27 + b + -18b2 = 0 Solving 27 + b + -18b2 = 0 Begin completing the square. Divide all terms by -18 the coefficient of the squared term: Divide each side by '-18'. -1.5 + -0.05555555556b + b2 = 0 Move the constant term to the right: Add '1.5' to each side of the equation. -1.5 + -0.05555555556b + 1.5 + b2 = 0 + 1.5 Reorder the terms: -1.5 + 1.5 + -0.05555555556b + b2 = 0 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + -0.05555555556b + b2 = 0 + 1.5 -0.05555555556b + b2 = 0 + 1.5 Combine like terms: 0 + 1.5 = 1.5 -0.05555555556b + b2 = 1.5 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. -0.05555555556b + 0.25 + b2 = 1.5 + 0.25 Reorder the terms: 0.25 + -0.05555555556b + b2 = 1.5 + 0.25 Combine like terms: 1.5 + 0.25 = 1.75 0.25 + -0.05555555556b + b2 = 1.75 Factor a perfect square on the left side: (b + 0.5)(b + 0.5) = 1.75 Calculate the square root of the right side: 1.322875656 Break this problem into two subproblems by setting (b + 0.5) equal to 1.322875656 and -1.322875656.Subproblem 1
b + 0.5 = 1.322875656 Simplifying b + 0.5 = 1.322875656 Reorder the terms: 0.5 + b = 1.322875656 Solving 0.5 + b = 1.322875656 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 = 1.322875656 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + b = 1.322875656 + -0.5 b = 1.322875656 + -0.5 Combine like terms: 1.322875656 + -0.5 = 0.822875656 b = 0.822875656 Simplifying b = 0.822875656Subproblem 2
b + 0.5 = -1.322875656 Simplifying b + 0.5 = -1.322875656 Reorder the terms: 0.5 + b = -1.322875656 Solving 0.5 + b = -1.322875656 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 = -1.322875656 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + b = -1.322875656 + -0.5 b = -1.322875656 + -0.5 Combine like terms: -1.322875656 + -0.5 = -1.822875656 b = -1.822875656 Simplifying b = -1.822875656Solution
The solution to the problem is based on the solutions from the subproblems. b = {0.822875656, -1.822875656}Solution
b = {0.822875656, -1.822875656}
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