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Simplifying -9(b2 + 2b + -3) + b(2b + 5) + 7 = 0 Reorder the terms: -9(-3 + 2b + b2) + b(2b + 5) + 7 = 0 (-3 * -9 + 2b * -9 + b2 * -9) + b(2b + 5) + 7 = 0 (27 + -18b + -9b2) + b(2b + 5) + 7 = 0 Reorder the terms: 27 + -18b + -9b2 + b(5 + 2b) + 7 = 0 27 + -18b + -9b2 + (5 * b + 2b * b) + 7 = 0 27 + -18b + -9b2 + (5b + 2b2) + 7 = 0 Reorder the terms: 27 + 7 + -18b + 5b + -9b2 + 2b2 = 0 Combine like terms: 27 + 7 = 34 34 + -18b + 5b + -9b2 + 2b2 = 0 Combine like terms: -18b + 5b = -13b 34 + -13b + -9b2 + 2b2 = 0 Combine like terms: -9b2 + 2b2 = -7b2 34 + -13b + -7b2 = 0 Solving 34 + -13b + -7b2 = 0 Solving for variable 'b'. Begin completing the square. Divide all terms by -7 the coefficient of the squared term: Divide each side by '-7'. -4.857142857 + 1.857142857b + b2 = 0 Move the constant term to the right: Add '4.857142857' to each side of the equation. -4.857142857 + 1.857142857b + 4.857142857 + b2 = 0 + 4.857142857 Reorder the terms: -4.857142857 + 4.857142857 + 1.857142857b + b2 = 0 + 4.857142857 Combine like terms: -4.857142857 + 4.857142857 = 0.000000000 0.000000000 + 1.857142857b + b2 = 0 + 4.857142857 1.857142857b + b2 = 0 + 4.857142857 Combine like terms: 0 + 4.857142857 = 4.857142857 1.857142857b + b2 = 4.857142857 The b term is 1.857142857b. Take half its coefficient (0.9285714285). Square it (0.8622448978) and add it to both sides. Add '0.8622448978' to each side of the equation. 1.857142857b + 0.8622448978 + b2 = 4.857142857 + 0.8622448978 Reorder the terms: 0.8622448978 + 1.857142857b + b2 = 4.857142857 + 0.8622448978 Combine like terms: 4.857142857 + 0.8622448978 = 5.7193877548 0.8622448978 + 1.857142857b + b2 = 5.7193877548 Factor a perfect square on the left side: (b + 0.9285714285)(b + 0.9285714285) = 5.7193877548 Calculate the square root of the right side: 2.391524149 Break this problem into two subproblems by setting (b + 0.9285714285) equal to 2.391524149 and -2.391524149.Subproblem 1
b + 0.9285714285 = 2.391524149 Simplifying b + 0.9285714285 = 2.391524149 Reorder the terms: 0.9285714285 + b = 2.391524149 Solving 0.9285714285 + b = 2.391524149 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-0.9285714285' to each side of the equation. 0.9285714285 + -0.9285714285 + b = 2.391524149 + -0.9285714285 Combine like terms: 0.9285714285 + -0.9285714285 = 0.0000000000 0.0000000000 + b = 2.391524149 + -0.9285714285 b = 2.391524149 + -0.9285714285 Combine like terms: 2.391524149 + -0.9285714285 = 1.4629527205 b = 1.4629527205 Simplifying b = 1.4629527205Subproblem 2
b + 0.9285714285 = -2.391524149 Simplifying b + 0.9285714285 = -2.391524149 Reorder the terms: 0.9285714285 + b = -2.391524149 Solving 0.9285714285 + b = -2.391524149 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-0.9285714285' to each side of the equation. 0.9285714285 + -0.9285714285 + b = -2.391524149 + -0.9285714285 Combine like terms: 0.9285714285 + -0.9285714285 = 0.0000000000 0.0000000000 + b = -2.391524149 + -0.9285714285 b = -2.391524149 + -0.9285714285 Combine like terms: -2.391524149 + -0.9285714285 = -3.3200955775 b = -3.3200955775 Simplifying b = -3.3200955775Solution
The solution to the problem is based on the solutions from the subproblems. b = {1.4629527205, -3.3200955775}
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