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Simplifying 12b2 + -8 = 6b Reorder the terms: -8 + 12b2 = 6b Solving -8 + 12b2 = 6b Solving for variable 'b'. Reorder the terms: -8 + -6b + 12b2 = 6b + -6b Combine like terms: 6b + -6b = 0 -8 + -6b + 12b2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-4 + -3b + 6b2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-4 + -3b + 6b2)' equal to zero and attempt to solve: Simplifying -4 + -3b + 6b2 = 0 Solving -4 + -3b + 6b2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -0.6666666667 + -0.5b + b2 = 0 Move the constant term to the right: Add '0.6666666667' to each side of the equation. -0.6666666667 + -0.5b + 0.6666666667 + b2 = 0 + 0.6666666667 Reorder the terms: -0.6666666667 + 0.6666666667 + -0.5b + b2 = 0 + 0.6666666667 Combine like terms: -0.6666666667 + 0.6666666667 = 0.0000000000 0.0000000000 + -0.5b + b2 = 0 + 0.6666666667 -0.5b + b2 = 0 + 0.6666666667 Combine like terms: 0 + 0.6666666667 = 0.6666666667 -0.5b + b2 = 0.6666666667 The b term is -0.5b. Take half its coefficient (-0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. -0.5b + 0.0625 + b2 = 0.6666666667 + 0.0625 Reorder the terms: 0.0625 + -0.5b + b2 = 0.6666666667 + 0.0625 Combine like terms: 0.6666666667 + 0.0625 = 0.7291666667 0.0625 + -0.5b + b2 = 0.7291666667 Factor a perfect square on the left side: (b + -0.25)(b + -0.25) = 0.7291666667 Calculate the square root of the right side: 0.853912564 Break this problem into two subproblems by setting (b + -0.25) equal to 0.853912564 and -0.853912564.Subproblem 1
b + -0.25 = 0.853912564 Simplifying b + -0.25 = 0.853912564 Reorder the terms: -0.25 + b = 0.853912564 Solving -0.25 + b = 0.853912564 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + b = 0.853912564 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + b = 0.853912564 + 0.25 b = 0.853912564 + 0.25 Combine like terms: 0.853912564 + 0.25 = 1.103912564 b = 1.103912564 Simplifying b = 1.103912564Subproblem 2
b + -0.25 = -0.853912564 Simplifying b + -0.25 = -0.853912564 Reorder the terms: -0.25 + b = -0.853912564 Solving -0.25 + b = -0.853912564 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '0.25' to each side of the equation. -0.25 + 0.25 + b = -0.853912564 + 0.25 Combine like terms: -0.25 + 0.25 = 0.00 0.00 + b = -0.853912564 + 0.25 b = -0.853912564 + 0.25 Combine like terms: -0.853912564 + 0.25 = -0.603912564 b = -0.603912564 Simplifying b = -0.603912564Solution
The solution to the problem is based on the solutions from the subproblems. b = {1.103912564, -0.603912564}Solution
b = {1.103912564, -0.603912564}
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