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Simplifying 6b2 + 3b + -14 = 0 Reorder the terms: -14 + 3b + 6b2 = 0 Solving -14 + 3b + 6b2 = 0 Solving for variable 'b'. Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -2.333333333 + 0.5b + b2 = 0 Move the constant term to the right: Add '2.333333333' to each side of the equation. -2.333333333 + 0.5b + 2.333333333 + b2 = 0 + 2.333333333 Reorder the terms: -2.333333333 + 2.333333333 + 0.5b + b2 = 0 + 2.333333333 Combine like terms: -2.333333333 + 2.333333333 = 0.000000000 0.000000000 + 0.5b + b2 = 0 + 2.333333333 0.5b + b2 = 0 + 2.333333333 Combine like terms: 0 + 2.333333333 = 2.333333333 0.5b + b2 = 2.333333333 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 = 2.333333333 + 0.0625 Reorder the terms: 0.0625 + 0.5b + b2 = 2.333333333 + 0.0625 Combine like terms: 2.333333333 + 0.0625 = 2.395833333 0.0625 + 0.5b + b2 = 2.395833333 Factor a perfect square on the left side: (b + 0.25)(b + 0.25) = 2.395833333 Calculate the square root of the right side: 1.547847968 Break this problem into two subproblems by setting (b + 0.25) equal to 1.547847968 and -1.547847968.Subproblem 1
b + 0.25 = 1.547847968 Simplifying b + 0.25 = 1.547847968 Reorder the terms: 0.25 + b = 1.547847968 Solving 0.25 + b = 1.547847968 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 = 1.547847968 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + b = 1.547847968 + -0.25 b = 1.547847968 + -0.25 Combine like terms: 1.547847968 + -0.25 = 1.297847968 b = 1.297847968 Simplifying b = 1.297847968Subproblem 2
b + 0.25 = -1.547847968 Simplifying b + 0.25 = -1.547847968 Reorder the terms: 0.25 + b = -1.547847968 Solving 0.25 + b = -1.547847968 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 = -1.547847968 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + b = -1.547847968 + -0.25 b = -1.547847968 + -0.25 Combine like terms: -1.547847968 + -0.25 = -1.797847968 b = -1.797847968 Simplifying b = -1.797847968Solution
The solution to the problem is based on the solutions from the subproblems. b = {1.297847968, -1.797847968}
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