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Simplifying 6b(b2 + 4b + -3) + 5b(-3b2 + b) = 0 Reorder the terms: 6b(-3 + 4b + b2) + 5b(-3b2 + b) = 0 (-3 * 6b + 4b * 6b + b2 * 6b) + 5b(-3b2 + b) = 0 (-18b + 24b2 + 6b3) + 5b(-3b2 + b) = 0 Reorder the terms: -18b + 24b2 + 6b3 + 5b(b + -3b2) = 0 -18b + 24b2 + 6b3 + (b * 5b + -3b2 * 5b) = 0 -18b + 24b2 + 6b3 + (5b2 + -15b3) = 0 Reorder the terms: -18b + 24b2 + 5b2 + 6b3 + -15b3 = 0 Combine like terms: 24b2 + 5b2 = 29b2 -18b + 29b2 + 6b3 + -15b3 = 0 Combine like terms: 6b3 + -15b3 = -9b3 -18b + 29b2 + -9b3 = 0 Solving -18b + 29b2 + -9b3 = 0 Solving for variable 'b'. Factor out the Greatest Common Factor (GCF), 'b'. b(-18 + 29b + -9b2) = 0Subproblem 1
Set the factor 'b' equal to zero and attempt to solve: Simplifying b = 0 Solving b = 0 Move all terms containing b to the left, all other terms to the right. Simplifying b = 0Subproblem 2
Set the factor '(-18 + 29b + -9b2)' equal to zero and attempt to solve: Simplifying -18 + 29b + -9b2 = 0 Solving -18 + 29b + -9b2 = 0 Begin completing the square. Divide all terms by -9 the coefficient of the squared term: Divide each side by '-9'. 2 + -3.222222222b + b2 = 0 Move the constant term to the right: Add '-2' to each side of the equation. 2 + -3.222222222b + -2 + b2 = 0 + -2 Reorder the terms: 2 + -2 + -3.222222222b + b2 = 0 + -2 Combine like terms: 2 + -2 = 0 0 + -3.222222222b + b2 = 0 + -2 -3.222222222b + b2 = 0 + -2 Combine like terms: 0 + -2 = -2 -3.222222222b + b2 = -2 The b term is -3.222222222b. Take half its coefficient (-1.611111111). Square it (2.595679012) and add it to both sides. Add '2.595679012' to each side of the equation. -3.222222222b + 2.595679012 + b2 = -2 + 2.595679012 Reorder the terms: 2.595679012 + -3.222222222b + b2 = -2 + 2.595679012 Combine like terms: -2 + 2.595679012 = 0.595679012 2.595679012 + -3.222222222b + b2 = 0.595679012 Factor a perfect square on the left side: (b + -1.611111111)(b + -1.611111111) = 0.595679012 Calculate the square root of the right side: 0.771802444 Break this problem into two subproblems by setting (b + -1.611111111) equal to 0.771802444 and -0.771802444.Subproblem 1
b + -1.611111111 = 0.771802444 Simplifying b + -1.611111111 = 0.771802444 Reorder the terms: -1.611111111 + b = 0.771802444 Solving -1.611111111 + b = 0.771802444 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '1.611111111' to each side of the equation. -1.611111111 + 1.611111111 + b = 0.771802444 + 1.611111111 Combine like terms: -1.611111111 + 1.611111111 = 0.000000000 0.000000000 + b = 0.771802444 + 1.611111111 b = 0.771802444 + 1.611111111 Combine like terms: 0.771802444 + 1.611111111 = 2.382913555 b = 2.382913555 Simplifying b = 2.382913555Subproblem 2
b + -1.611111111 = -0.771802444 Simplifying b + -1.611111111 = -0.771802444 Reorder the terms: -1.611111111 + b = -0.771802444 Solving -1.611111111 + b = -0.771802444 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '1.611111111' to each side of the equation. -1.611111111 + 1.611111111 + b = -0.771802444 + 1.611111111 Combine like terms: -1.611111111 + 1.611111111 = 0.000000000 0.000000000 + b = -0.771802444 + 1.611111111 b = -0.771802444 + 1.611111111 Combine like terms: -0.771802444 + 1.611111111 = 0.839308667 b = 0.839308667 Simplifying b = 0.839308667Solution
The solution to the problem is based on the solutions from the subproblems. b = {2.382913555, 0.839308667}Solution
b = {0, 2.382913555, 0.839308667}
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