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Simplifying -9b(b + 1) + 4 = 3(2b + -7) + -1b Reorder the terms: -9b(1 + b) + 4 = 3(2b + -7) + -1b (1 * -9b + b * -9b) + 4 = 3(2b + -7) + -1b (-9b + -9b2) + 4 = 3(2b + -7) + -1b Reorder the terms: 4 + -9b + -9b2 = 3(2b + -7) + -1b Reorder the terms: 4 + -9b + -9b2 = 3(-7 + 2b) + -1b 4 + -9b + -9b2 = (-7 * 3 + 2b * 3) + -1b 4 + -9b + -9b2 = (-21 + 6b) + -1b Combine like terms: 6b + -1b = 5b 4 + -9b + -9b2 = -21 + 5b Solving 4 + -9b + -9b2 = -21 + 5b Solving for variable 'b'. Reorder the terms: 4 + 21 + -9b + -5b + -9b2 = -21 + 5b + 21 + -5b Combine like terms: 4 + 21 = 25 25 + -9b + -5b + -9b2 = -21 + 5b + 21 + -5b Combine like terms: -9b + -5b = -14b 25 + -14b + -9b2 = -21 + 5b + 21 + -5b Reorder the terms: 25 + -14b + -9b2 = -21 + 21 + 5b + -5b Combine like terms: -21 + 21 = 0 25 + -14b + -9b2 = 0 + 5b + -5b 25 + -14b + -9b2 = 5b + -5b Combine like terms: 5b + -5b = 0 25 + -14b + -9b2 = 0 Begin completing the square. Divide all terms by -9 the coefficient of the squared term: Divide each side by '-9'. -2.777777778 + 1.555555556b + b2 = 0 Move the constant term to the right: Add '2.777777778' to each side of the equation. -2.777777778 + 1.555555556b + 2.777777778 + b2 = 0 + 2.777777778 Reorder the terms: -2.777777778 + 2.777777778 + 1.555555556b + b2 = 0 + 2.777777778 Combine like terms: -2.777777778 + 2.777777778 = 0.000000000 0.000000000 + 1.555555556b + b2 = 0 + 2.777777778 1.555555556b + b2 = 0 + 2.777777778 Combine like terms: 0 + 2.777777778 = 2.777777778 1.555555556b + b2 = 2.777777778 The b term is 1.555555556b. Take half its coefficient (0.777777778). Square it (0.6049382720) and add it to both sides. Add '0.6049382720' to each side of the equation. 1.555555556b + 0.6049382720 + b2 = 2.777777778 + 0.6049382720 Reorder the terms: 0.6049382720 + 1.555555556b + b2 = 2.777777778 + 0.6049382720 Combine like terms: 2.777777778 + 0.6049382720 = 3.38271605 0.6049382720 + 1.555555556b + b2 = 3.38271605 Factor a perfect square on the left side: (b + 0.777777778)(b + 0.777777778) = 3.38271605 Calculate the square root of the right side: 1.839216151 Break this problem into two subproblems by setting (b + 0.777777778) equal to 1.839216151 and -1.839216151.Subproblem 1
b + 0.777777778 = 1.839216151 Simplifying b + 0.777777778 = 1.839216151 Reorder the terms: 0.777777778 + b = 1.839216151 Solving 0.777777778 + b = 1.839216151 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-0.777777778' to each side of the equation. 0.777777778 + -0.777777778 + b = 1.839216151 + -0.777777778 Combine like terms: 0.777777778 + -0.777777778 = 0.000000000 0.000000000 + b = 1.839216151 + -0.777777778 b = 1.839216151 + -0.777777778 Combine like terms: 1.839216151 + -0.777777778 = 1.061438373 b = 1.061438373 Simplifying b = 1.061438373Subproblem 2
b + 0.777777778 = -1.839216151 Simplifying b + 0.777777778 = -1.839216151 Reorder the terms: 0.777777778 + b = -1.839216151 Solving 0.777777778 + b = -1.839216151 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-0.777777778' to each side of the equation. 0.777777778 + -0.777777778 + b = -1.839216151 + -0.777777778 Combine like terms: 0.777777778 + -0.777777778 = 0.000000000 0.000000000 + b = -1.839216151 + -0.777777778 b = -1.839216151 + -0.777777778 Combine like terms: -1.839216151 + -0.777777778 = -2.616993929 b = -2.616993929 Simplifying b = -2.616993929Solution
The solution to the problem is based on the solutions from the subproblems. b = {1.061438373, -2.616993929}
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