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Simplifying 6b2 + -3b + 4 + -12b2 + -3b + 5 = 0 Reorder the terms: 4 + 5 + -3b + -3b + 6b2 + -12b2 = 0 Combine like terms: 4 + 5 = 9 9 + -3b + -3b + 6b2 + -12b2 = 0 Combine like terms: -3b + -3b = -6b 9 + -6b + 6b2 + -12b2 = 0 Combine like terms: 6b2 + -12b2 = -6b2 9 + -6b + -6b2 = 0 Solving 9 + -6b + -6b2 = 0 Solving for variable 'b'. Factor out the Greatest Common Factor (GCF), '3'. 3(3 + -2b + -2b2) = 0 Ignore the factor 3.Subproblem 1
Set the factor '(3 + -2b + -2b2)' equal to zero and attempt to solve: Simplifying 3 + -2b + -2b2 = 0 Solving 3 + -2b + -2b2 = 0 Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -1.5 + b + b2 = 0 Move the constant term to the right: Add '1.5' to each side of the equation. -1.5 + b + 1.5 + b2 = 0 + 1.5 Reorder the terms: -1.5 + 1.5 + b + b2 = 0 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + b + b2 = 0 + 1.5 b + b2 = 0 + 1.5 Combine like terms: 0 + 1.5 = 1.5 b + b2 = 1.5 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. + 0.25 + b2 = 1.5 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + b2 = 1.5 + 0.25 Combine like terms: 1.5 + 0.25 = 1.75 1.25 + b2 = 1.75 Factor a perfect square on the left side: (b + 0.5)(b + 0.5) = 1.75 Calculate the square root of the right side: 1.322875656 Break this problem into two subproblems by setting (b + 0.5) equal to 1.322875656 and -1.322875656.Subproblem 1
b + 0.5 = 1.322875656 Simplifying b + 0.5 = 1.322875656 Reorder the terms: 0.5 + b = 1.322875656 Solving 0.5 + b = 1.322875656 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 = 1.322875656 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + b = 1.322875656 + -0.5 b = 1.322875656 + -0.5 Combine like terms: 1.322875656 + -0.5 = 0.822875656 b = 0.822875656 Simplifying b = 0.822875656Subproblem 2
b + 0.5 = -1.322875656 Simplifying b + 0.5 = -1.322875656 Reorder the terms: 0.5 + b = -1.322875656 Solving 0.5 + b = -1.322875656 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 = -1.322875656 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + b = -1.322875656 + -0.5 b = -1.322875656 + -0.5 Combine like terms: -1.322875656 + -0.5 = -1.822875656 b = -1.822875656 Simplifying b = -1.822875656Solution
The solution to the problem is based on the solutions from the subproblems. b = {0.822875656, -1.822875656}Solution
b = {0.822875656, -1.822875656}
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