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Simplifying 4b(-5b + -3) + -2(b2 + -7b + -4) = 0 Reorder the terms: 4b(-3 + -5b) + -2(b2 + -7b + -4) = 0 (-3 * 4b + -5b * 4b) + -2(b2 + -7b + -4) = 0 (-12b + -20b2) + -2(b2 + -7b + -4) = 0 Reorder the terms: -12b + -20b2 + -2(-4 + -7b + b2) = 0 -12b + -20b2 + (-4 * -2 + -7b * -2 + b2 * -2) = 0 -12b + -20b2 + (8 + 14b + -2b2) = 0 Reorder the terms: 8 + -12b + 14b + -20b2 + -2b2 = 0 Combine like terms: -12b + 14b = 2b 8 + 2b + -20b2 + -2b2 = 0 Combine like terms: -20b2 + -2b2 = -22b2 8 + 2b + -22b2 = 0 Solving 8 + 2b + -22b2 = 0 Solving for variable 'b'. Factor out the Greatest Common Factor (GCF), '2'. 2(4 + b + -11b2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(4 + b + -11b2)' equal to zero and attempt to solve: Simplifying 4 + b + -11b2 = 0 Solving 4 + b + -11b2 = 0 Begin completing the square. Divide all terms by -11 the coefficient of the squared term: Divide each side by '-11'. -0.3636363636 + -0.09090909091b + b2 = 0 Move the constant term to the right: Add '0.3636363636' to each side of the equation. -0.3636363636 + -0.09090909091b + 0.3636363636 + b2 = 0 + 0.3636363636 Reorder the terms: -0.3636363636 + 0.3636363636 + -0.09090909091b + b2 = 0 + 0.3636363636 Combine like terms: -0.3636363636 + 0.3636363636 = 0.0000000000 0.0000000000 + -0.09090909091b + b2 = 0 + 0.3636363636 -0.09090909091b + b2 = 0 + 0.3636363636 Combine like terms: 0 + 0.3636363636 = 0.3636363636 -0.09090909091b + b2 = 0.3636363636 The b term is b. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. -0.09090909091b + 0.25 + b2 = 0.3636363636 + 0.25 Reorder the terms: 0.25 + -0.09090909091b + b2 = 0.3636363636 + 0.25 Combine like terms: 0.3636363636 + 0.25 = 0.6136363636 0.25 + -0.09090909091b + b2 = 0.6136363636 Factor a perfect square on the left side: (b + 0.5)(b + 0.5) = 0.6136363636 Calculate the square root of the right side: 0.783349452 Break this problem into two subproblems by setting (b + 0.5) equal to 0.783349452 and -0.783349452.Subproblem 1
b + 0.5 = 0.783349452 Simplifying b + 0.5 = 0.783349452 Reorder the terms: 0.5 + b = 0.783349452 Solving 0.5 + b = 0.783349452 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + b = 0.783349452 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + b = 0.783349452 + -0.5 b = 0.783349452 + -0.5 Combine like terms: 0.783349452 + -0.5 = 0.283349452 b = 0.283349452 Simplifying b = 0.283349452Subproblem 2
b + 0.5 = -0.783349452 Simplifying b + 0.5 = -0.783349452 Reorder the terms: 0.5 + b = -0.783349452 Solving 0.5 + b = -0.783349452 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + b = -0.783349452 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + b = -0.783349452 + -0.5 b = -0.783349452 + -0.5 Combine like terms: -0.783349452 + -0.5 = -1.283349452 b = -1.283349452 Simplifying b = -1.283349452Solution
The solution to the problem is based on the solutions from the subproblems. b = {0.283349452, -1.283349452}Solution
b = {0.283349452, -1.283349452}
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