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Simplifying 4b2 + -4b + -27 = 0 Reorder the terms: -27 + -4b + 4b2 = 0 Solving -27 + -4b + 4b2 = 0 Solving for variable 'b'. Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -6.75 + -1b + b2 = 0 Move the constant term to the right: Add '6.75' to each side of the equation. -6.75 + -1b + 6.75 + b2 = 0 + 6.75 Reorder the terms: -6.75 + 6.75 + -1b + b2 = 0 + 6.75 Combine like terms: -6.75 + 6.75 = 0.00 0.00 + -1b + b2 = 0 + 6.75 -1b + b2 = 0 + 6.75 Combine like terms: 0 + 6.75 = 6.75 -1b + b2 = 6.75 The b term is -1b. 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. -1b + 0.25 + b2 = 6.75 + 0.25 Reorder the terms: 0.25 + -1b + b2 = 6.75 + 0.25 Combine like terms: 6.75 + 0.25 = 7 0.25 + -1b + b2 = 7 Factor a perfect square on the left side: (b + -0.5)(b + -0.5) = 7 Calculate the square root of the right side: 2.645751311 Break this problem into two subproblems by setting (b + -0.5) equal to 2.645751311 and -2.645751311.Subproblem 1
b + -0.5 = 2.645751311 Simplifying b + -0.5 = 2.645751311 Reorder the terms: -0.5 + b = 2.645751311 Solving -0.5 + b = 2.645751311 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 = 2.645751311 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + b = 2.645751311 + 0.5 b = 2.645751311 + 0.5 Combine like terms: 2.645751311 + 0.5 = 3.145751311 b = 3.145751311 Simplifying b = 3.145751311Subproblem 2
b + -0.5 = -2.645751311 Simplifying b + -0.5 = -2.645751311 Reorder the terms: -0.5 + b = -2.645751311 Solving -0.5 + b = -2.645751311 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 = -2.645751311 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + b = -2.645751311 + 0.5 b = -2.645751311 + 0.5 Combine like terms: -2.645751311 + 0.5 = -2.145751311 b = -2.145751311 Simplifying b = -2.145751311Solution
The solution to the problem is based on the solutions from the subproblems. b = {3.145751311, -2.145751311}
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