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Simplifying 6b2 + 12b + -78 = 6 Reorder the terms: -78 + 12b + 6b2 = 6 Solving -78 + 12b + 6b2 = 6 Solving for variable 'b'. Reorder the terms: -78 + -6 + 12b + 6b2 = 6 + -6 Combine like terms: -78 + -6 = -84 -84 + 12b + 6b2 = 6 + -6 Combine like terms: 6 + -6 = 0 -84 + 12b + 6b2 = 0 Factor out the Greatest Common Factor (GCF), '6'. 6(-14 + 2b + b2) = 0 Ignore the factor 6.Subproblem 1
Set the factor '(-14 + 2b + b2)' equal to zero and attempt to solve: Simplifying -14 + 2b + b2 = 0 Solving -14 + 2b + b2 = 0 Begin completing the square. Move the constant term to the right: Add '14' to each side of the equation. -14 + 2b + 14 + b2 = 0 + 14 Reorder the terms: -14 + 14 + 2b + b2 = 0 + 14 Combine like terms: -14 + 14 = 0 0 + 2b + b2 = 0 + 14 2b + b2 = 0 + 14 Combine like terms: 0 + 14 = 14 2b + b2 = 14 The b term is 2b. Take half its coefficient (1). Square it (1) and add it to both sides. Add '1' to each side of the equation. 2b + 1 + b2 = 14 + 1 Reorder the terms: 1 + 2b + b2 = 14 + 1 Combine like terms: 14 + 1 = 15 1 + 2b + b2 = 15 Factor a perfect square on the left side: (b + 1)(b + 1) = 15 Calculate the square root of the right side: 3.872983346 Break this problem into two subproblems by setting (b + 1) equal to 3.872983346 and -3.872983346.Subproblem 1
b + 1 = 3.872983346 Simplifying b + 1 = 3.872983346 Reorder the terms: 1 + b = 3.872983346 Solving 1 + b = 3.872983346 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + b = 3.872983346 + -1 Combine like terms: 1 + -1 = 0 0 + b = 3.872983346 + -1 b = 3.872983346 + -1 Combine like terms: 3.872983346 + -1 = 2.872983346 b = 2.872983346 Simplifying b = 2.872983346Subproblem 2
b + 1 = -3.872983346 Simplifying b + 1 = -3.872983346 Reorder the terms: 1 + b = -3.872983346 Solving 1 + b = -3.872983346 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + b = -3.872983346 + -1 Combine like terms: 1 + -1 = 0 0 + b = -3.872983346 + -1 b = -3.872983346 + -1 Combine like terms: -3.872983346 + -1 = -4.872983346 b = -4.872983346 Simplifying b = -4.872983346Solution
The solution to the problem is based on the solutions from the subproblems. b = {2.872983346, -4.872983346}Solution
b = {2.872983346, -4.872983346}
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