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Simplifying 2(11.5 + 1.5b) * 3b = 29 Reorder the terms for easier multiplication: 2 * 3b(11.5 + 1.5b) = 29 Multiply 2 * 3 6b(11.5 + 1.5b) = 29 (11.5 * 6b + 1.5b * 6b) = 29 (69b + 9b2) = 29 Solving 69b + 9b2 = 29 Solving for variable 'b'. Reorder the terms: -29 + 69b + 9b2 = 29 + -29 Combine like terms: 29 + -29 = 0 -29 + 69b + 9b2 = 0 Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. -3.222222222 + 7.666666667b + b2 = 0 Move the constant term to the right: Add '3.222222222' to each side of the equation. -3.222222222 + 7.666666667b + 3.222222222 + b2 = 0 + 3.222222222 Reorder the terms: -3.222222222 + 3.222222222 + 7.666666667b + b2 = 0 + 3.222222222 Combine like terms: -3.222222222 + 3.222222222 = 0.000000000 0.000000000 + 7.666666667b + b2 = 0 + 3.222222222 7.666666667b + b2 = 0 + 3.222222222 Combine like terms: 0 + 3.222222222 = 3.222222222 7.666666667b + b2 = 3.222222222 The b term is 7.666666667b. Take half its coefficient (3.833333334). Square it (14.69444445) and add it to both sides. Add '14.69444445' to each side of the equation. 7.666666667b + 14.69444445 + b2 = 3.222222222 + 14.69444445 Reorder the terms: 14.69444445 + 7.666666667b + b2 = 3.222222222 + 14.69444445 Combine like terms: 3.222222222 + 14.69444445 = 17.916666672 14.69444445 + 7.666666667b + b2 = 17.916666672 Factor a perfect square on the left side: (b + 3.833333334)(b + 3.833333334) = 17.916666672 Calculate the square root of the right side: 4.232808367 Break this problem into two subproblems by setting (b + 3.833333334) equal to 4.232808367 and -4.232808367.Subproblem 1
b + 3.833333334 = 4.232808367 Simplifying b + 3.833333334 = 4.232808367 Reorder the terms: 3.833333334 + b = 4.232808367 Solving 3.833333334 + b = 4.232808367 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-3.833333334' to each side of the equation. 3.833333334 + -3.833333334 + b = 4.232808367 + -3.833333334 Combine like terms: 3.833333334 + -3.833333334 = 0.000000000 0.000000000 + b = 4.232808367 + -3.833333334 b = 4.232808367 + -3.833333334 Combine like terms: 4.232808367 + -3.833333334 = 0.399475033 b = 0.399475033 Simplifying b = 0.399475033Subproblem 2
b + 3.833333334 = -4.232808367 Simplifying b + 3.833333334 = -4.232808367 Reorder the terms: 3.833333334 + b = -4.232808367 Solving 3.833333334 + b = -4.232808367 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-3.833333334' to each side of the equation. 3.833333334 + -3.833333334 + b = -4.232808367 + -3.833333334 Combine like terms: 3.833333334 + -3.833333334 = 0.000000000 0.000000000 + b = -4.232808367 + -3.833333334 b = -4.232808367 + -3.833333334 Combine like terms: -4.232808367 + -3.833333334 = -8.066141701 b = -8.066141701 Simplifying b = -8.066141701Solution
The solution to the problem is based on the solutions from the subproblems. b = {0.399475033, -8.066141701}
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