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Simplifying 10b2 + 30b + -27 = 0 Reorder the terms: -27 + 30b + 10b2 = 0 Solving -27 + 30b + 10b2 = 0 Solving for variable 'b'. Begin completing the square. Divide all terms by 10 the coefficient of the squared term: Divide each side by '10'. -2.7 + 3b + b2 = 0 Move the constant term to the right: Add '2.7' to each side of the equation. -2.7 + 3b + 2.7 + b2 = 0 + 2.7 Reorder the terms: -2.7 + 2.7 + 3b + b2 = 0 + 2.7 Combine like terms: -2.7 + 2.7 = 0.0 0.0 + 3b + b2 = 0 + 2.7 3b + b2 = 0 + 2.7 Combine like terms: 0 + 2.7 = 2.7 3b + b2 = 2.7 The b term is 3b. Take half its coefficient (1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. 3b + 2.25 + b2 = 2.7 + 2.25 Reorder the terms: 2.25 + 3b + b2 = 2.7 + 2.25 Combine like terms: 2.7 + 2.25 = 4.95 2.25 + 3b + b2 = 4.95 Factor a perfect square on the left side: (b + 1.5)(b + 1.5) = 4.95 Calculate the square root of the right side: 2.224859546 Break this problem into two subproblems by setting (b + 1.5) equal to 2.224859546 and -2.224859546.Subproblem 1
b + 1.5 = 2.224859546 Simplifying b + 1.5 = 2.224859546 Reorder the terms: 1.5 + b = 2.224859546 Solving 1.5 + b = 2.224859546 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + b = 2.224859546 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + b = 2.224859546 + -1.5 b = 2.224859546 + -1.5 Combine like terms: 2.224859546 + -1.5 = 0.724859546 b = 0.724859546 Simplifying b = 0.724859546Subproblem 2
b + 1.5 = -2.224859546 Simplifying b + 1.5 = -2.224859546 Reorder the terms: 1.5 + b = -2.224859546 Solving 1.5 + b = -2.224859546 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-1.5' to each side of the equation. 1.5 + -1.5 + b = -2.224859546 + -1.5 Combine like terms: 1.5 + -1.5 = 0.0 0.0 + b = -2.224859546 + -1.5 b = -2.224859546 + -1.5 Combine like terms: -2.224859546 + -1.5 = -3.724859546 b = -3.724859546 Simplifying b = -3.724859546Solution
The solution to the problem is based on the solutions from the subproblems. b = {0.724859546, -3.724859546}
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