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Simplifying 1.5b(b + 2) = 90 Reorder the terms: 1.5b(2 + b) = 90 (2 * 1.5b + b * 1.5b) = 90 (3b + 1.5b2) = 90 Solving 3b + 1.5b2 = 90 Solving for variable 'b'. Reorder the terms: -90 + 3b + 1.5b2 = 90 + -90 Combine like terms: 90 + -90 = 0 -90 + 3b + 1.5b2 = 0 Begin completing the square. Divide all terms by 1.5 the coefficient of the squared term: Divide each side by '1.5'. -60 + 2b + b2 = 0 Move the constant term to the right: Add '60' to each side of the equation. -60 + 2b + 60 + b2 = 0 + 60 Reorder the terms: -60 + 60 + 2b + b2 = 0 + 60 Combine like terms: -60 + 60 = 0 0 + 2b + b2 = 0 + 60 2b + b2 = 0 + 60 Combine like terms: 0 + 60 = 60 2b + b2 = 60 The b term is 2b. Take half its coefficient (1). Square it (1) and add it to both sides. Add '1' to each side of the equation. 2b + 1 + b2 = 60 + 1 Reorder the terms: 1 + 2b + b2 = 60 + 1 Combine like terms: 60 + 1 = 61 1 + 2b + b2 = 61 Factor a perfect square on the left side: (b + 1)(b + 1) = 61 Calculate the square root of the right side: 7.810249676 Break this problem into two subproblems by setting (b + 1) equal to 7.810249676 and -7.810249676.Subproblem 1
b + 1 = 7.810249676 Simplifying b + 1 = 7.810249676 Reorder the terms: 1 + b = 7.810249676 Solving 1 + b = 7.810249676 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + b = 7.810249676 + -1 Combine like terms: 1 + -1 = 0 0 + b = 7.810249676 + -1 b = 7.810249676 + -1 Combine like terms: 7.810249676 + -1 = 6.810249676 b = 6.810249676 Simplifying b = 6.810249676Subproblem 2
b + 1 = -7.810249676 Simplifying b + 1 = -7.810249676 Reorder the terms: 1 + b = -7.810249676 Solving 1 + b = -7.810249676 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + b = -7.810249676 + -1 Combine like terms: 1 + -1 = 0 0 + b = -7.810249676 + -1 b = -7.810249676 + -1 Combine like terms: -7.810249676 + -1 = -8.810249676 b = -8.810249676 Simplifying b = -8.810249676Solution
The solution to the problem is based on the solutions from the subproblems. b = {6.810249676, -8.810249676}
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