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Simplifying 3b(6b2 + -5b + -7) = 0 Reorder the terms: 3b(-7 + -5b + 6b2) = 0 (-7 * 3b + -5b * 3b + 6b2 * 3b) = 0 (-21b + -15b2 + 18b3) = 0 Solving -21b + -15b2 + 18b3 = 0 Solving for variable 'b'. Factor out the Greatest Common Factor (GCF), '3b'. 3b(-7 + -5b + 6b2) = 0 Ignore the factor 3.Subproblem 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 '(-7 + -5b + 6b2)' equal to zero and attempt to solve: Simplifying -7 + -5b + 6b2 = 0 Solving -7 + -5b + 6b2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -1.166666667 + -0.8333333333b + b2 = 0 Move the constant term to the right: Add '1.166666667' to each side of the equation. -1.166666667 + -0.8333333333b + 1.166666667 + b2 = 0 + 1.166666667 Reorder the terms: -1.166666667 + 1.166666667 + -0.8333333333b + b2 = 0 + 1.166666667 Combine like terms: -1.166666667 + 1.166666667 = 0.000000000 0.000000000 + -0.8333333333b + b2 = 0 + 1.166666667 -0.8333333333b + b2 = 0 + 1.166666667 Combine like terms: 0 + 1.166666667 = 1.166666667 -0.8333333333b + b2 = 1.166666667 The b term is -0.8333333333b. Take half its coefficient (-0.4166666667). Square it (0.1736111111) and add it to both sides. Add '0.1736111111' to each side of the equation. -0.8333333333b + 0.1736111111 + b2 = 1.166666667 + 0.1736111111 Reorder the terms: 0.1736111111 + -0.8333333333b + b2 = 1.166666667 + 0.1736111111 Combine like terms: 1.166666667 + 0.1736111111 = 1.3402777781 0.1736111111 + -0.8333333333b + b2 = 1.3402777781 Factor a perfect square on the left side: (b + -0.4166666667)(b + -0.4166666667) = 1.3402777781 Calculate the square root of the right side: 1.157703666 Break this problem into two subproblems by setting (b + -0.4166666667) equal to 1.157703666 and -1.157703666.Subproblem 1
b + -0.4166666667 = 1.157703666 Simplifying b + -0.4166666667 = 1.157703666 Reorder the terms: -0.4166666667 + b = 1.157703666 Solving -0.4166666667 + b = 1.157703666 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '0.4166666667' to each side of the equation. -0.4166666667 + 0.4166666667 + b = 1.157703666 + 0.4166666667 Combine like terms: -0.4166666667 + 0.4166666667 = 0.0000000000 0.0000000000 + b = 1.157703666 + 0.4166666667 b = 1.157703666 + 0.4166666667 Combine like terms: 1.157703666 + 0.4166666667 = 1.5743703327 b = 1.5743703327 Simplifying b = 1.5743703327Subproblem 2
b + -0.4166666667 = -1.157703666 Simplifying b + -0.4166666667 = -1.157703666 Reorder the terms: -0.4166666667 + b = -1.157703666 Solving -0.4166666667 + b = -1.157703666 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '0.4166666667' to each side of the equation. -0.4166666667 + 0.4166666667 + b = -1.157703666 + 0.4166666667 Combine like terms: -0.4166666667 + 0.4166666667 = 0.0000000000 0.0000000000 + b = -1.157703666 + 0.4166666667 b = -1.157703666 + 0.4166666667 Combine like terms: -1.157703666 + 0.4166666667 = -0.7410369993 b = -0.7410369993 Simplifying b = -0.7410369993Solution
The solution to the problem is based on the solutions from the subproblems. b = {1.5743703327, -0.7410369993}Solution
b = {0, 1.5743703327, -0.7410369993}
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