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Simplifying (5b3 + -7b + 5b4) + -1(8b2 + 5b4 + -5b) = 0 Reorder the terms: (-7b + 5b3 + 5b4) + -1(8b2 + 5b4 + -5b) = 0 Remove parenthesis around (-7b + 5b3 + 5b4) -7b + 5b3 + 5b4 + -1(8b2 + 5b4 + -5b) = 0 Reorder the terms: -7b + 5b3 + 5b4 + -1(-5b + 8b2 + 5b4) = 0 -7b + 5b3 + 5b4 + (-5b * -1 + 8b2 * -1 + 5b4 * -1) = 0 -7b + 5b3 + 5b4 + (5b + -8b2 + -5b4) = 0 Reorder the terms: -7b + 5b + -8b2 + 5b3 + 5b4 + -5b4 = 0 Combine like terms: -7b + 5b = -2b -2b + -8b2 + 5b3 + 5b4 + -5b4 = 0 Combine like terms: 5b4 + -5b4 = 0 -2b + -8b2 + 5b3 + 0 = 0 -2b + -8b2 + 5b3 = 0 Solving -2b + -8b2 + 5b3 = 0 Solving for variable 'b'. Factor out the Greatest Common Factor (GCF), 'b'. b(-2 + -8b + 5b2) = 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 '(-2 + -8b + 5b2)' equal to zero and attempt to solve: Simplifying -2 + -8b + 5b2 = 0 Solving -2 + -8b + 5b2 = 0 Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -0.4 + -1.6b + b2 = 0 Move the constant term to the right: Add '0.4' to each side of the equation. -0.4 + -1.6b + 0.4 + b2 = 0 + 0.4 Reorder the terms: -0.4 + 0.4 + -1.6b + b2 = 0 + 0.4 Combine like terms: -0.4 + 0.4 = 0.0 0.0 + -1.6b + b2 = 0 + 0.4 -1.6b + b2 = 0 + 0.4 Combine like terms: 0 + 0.4 = 0.4 -1.6b + b2 = 0.4 The b term is -1.6b. Take half its coefficient (-0.8). Square it (0.64) and add it to both sides. Add '0.64' to each side of the equation. -1.6b + 0.64 + b2 = 0.4 + 0.64 Reorder the terms: 0.64 + -1.6b + b2 = 0.4 + 0.64 Combine like terms: 0.4 + 0.64 = 1.04 0.64 + -1.6b + b2 = 1.04 Factor a perfect square on the left side: (b + -0.8)(b + -0.8) = 1.04 Calculate the square root of the right side: 1.019803903 Break this problem into two subproblems by setting (b + -0.8) equal to 1.019803903 and -1.019803903.Subproblem 1
b + -0.8 = 1.019803903 Simplifying b + -0.8 = 1.019803903 Reorder the terms: -0.8 + b = 1.019803903 Solving -0.8 + b = 1.019803903 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '0.8' to each side of the equation. -0.8 + 0.8 + b = 1.019803903 + 0.8 Combine like terms: -0.8 + 0.8 = 0.0 0.0 + b = 1.019803903 + 0.8 b = 1.019803903 + 0.8 Combine like terms: 1.019803903 + 0.8 = 1.819803903 b = 1.819803903 Simplifying b = 1.819803903Subproblem 2
b + -0.8 = -1.019803903 Simplifying b + -0.8 = -1.019803903 Reorder the terms: -0.8 + b = -1.019803903 Solving -0.8 + b = -1.019803903 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '0.8' to each side of the equation. -0.8 + 0.8 + b = -1.019803903 + 0.8 Combine like terms: -0.8 + 0.8 = 0.0 0.0 + b = -1.019803903 + 0.8 b = -1.019803903 + 0.8 Combine like terms: -1.019803903 + 0.8 = -0.219803903 b = -0.219803903 Simplifying b = -0.219803903Solution
The solution to the problem is based on the solutions from the subproblems. b = {1.819803903, -0.219803903}Solution
b = {0, 1.819803903, -0.219803903}
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