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Simplifying 1.5b(b + 2) = 2b + -9 + 15 Reorder the terms: 1.5b(2 + b) = 2b + -9 + 15 (2 * 1.5b + b * 1.5b) = 2b + -9 + 15 (3b + 1.5b2) = 2b + -9 + 15 Reorder the terms: 3b + 1.5b2 = -9 + 15 + 2b Combine like terms: -9 + 15 = 6 3b + 1.5b2 = 6 + 2b Solving 3b + 1.5b2 = 6 + 2b Solving for variable 'b'. Reorder the terms: -6 + 3b + -2b + 1.5b2 = 6 + 2b + -6 + -2b Combine like terms: 3b + -2b = 1b -6 + 1b + 1.5b2 = 6 + 2b + -6 + -2b Reorder the terms: -6 + 1b + 1.5b2 = 6 + -6 + 2b + -2b Combine like terms: 6 + -6 = 0 -6 + 1b + 1.5b2 = 0 + 2b + -2b -6 + 1b + 1.5b2 = 2b + -2b Combine like terms: 2b + -2b = 0 -6 + 1b + 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'. -4 + 0.6666666667b + b2 = 0 Move the constant term to the right: Add '4' to each side of the equation. -4 + 0.6666666667b + 4 + b2 = 0 + 4 Reorder the terms: -4 + 4 + 0.6666666667b + b2 = 0 + 4 Combine like terms: -4 + 4 = 0 0 + 0.6666666667b + b2 = 0 + 4 0.6666666667b + b2 = 0 + 4 Combine like terms: 0 + 4 = 4 0.6666666667b + b2 = 4 The b term is 0.6666666667b. Take half its coefficient (0.3333333334). Square it (0.1111111112) and add it to both sides. Add '0.1111111112' to each side of the equation. 0.6666666667b + 0.1111111112 + b2 = 4 + 0.1111111112 Reorder the terms: 0.1111111112 + 0.6666666667b + b2 = 4 + 0.1111111112 Combine like terms: 4 + 0.1111111112 = 4.1111111112 0.1111111112 + 0.6666666667b + b2 = 4.1111111112 Factor a perfect square on the left side: (b + 0.3333333334)(b + 0.3333333334) = 4.1111111112 Calculate the square root of the right side: 2.02758751 Break this problem into two subproblems by setting (b + 0.3333333334) equal to 2.02758751 and -2.02758751.Subproblem 1
b + 0.3333333334 = 2.02758751 Simplifying b + 0.3333333334 = 2.02758751 Reorder the terms: 0.3333333334 + b = 2.02758751 Solving 0.3333333334 + b = 2.02758751 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-0.3333333334' to each side of the equation. 0.3333333334 + -0.3333333334 + b = 2.02758751 + -0.3333333334 Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000 0.0000000000 + b = 2.02758751 + -0.3333333334 b = 2.02758751 + -0.3333333334 Combine like terms: 2.02758751 + -0.3333333334 = 1.6942541766 b = 1.6942541766 Simplifying b = 1.6942541766Subproblem 2
b + 0.3333333334 = -2.02758751 Simplifying b + 0.3333333334 = -2.02758751 Reorder the terms: 0.3333333334 + b = -2.02758751 Solving 0.3333333334 + b = -2.02758751 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-0.3333333334' to each side of the equation. 0.3333333334 + -0.3333333334 + b = -2.02758751 + -0.3333333334 Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000 0.0000000000 + b = -2.02758751 + -0.3333333334 b = -2.02758751 + -0.3333333334 Combine like terms: -2.02758751 + -0.3333333334 = -2.3609208434 b = -2.3609208434 Simplifying b = -2.3609208434Solution
The solution to the problem is based on the solutions from the subproblems. b = {1.6942541766, -2.3609208434}
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