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Simplifying 2b2 = 6b + 7 Reorder the terms: 2b2 = 7 + 6b Solving 2b2 = 7 + 6b Solving for variable 'b'. Reorder the terms: -7 + -6b + 2b2 = 7 + 6b + -7 + -6b Reorder the terms: -7 + -6b + 2b2 = 7 + -7 + 6b + -6b Combine like terms: 7 + -7 = 0 -7 + -6b + 2b2 = 0 + 6b + -6b -7 + -6b + 2b2 = 6b + -6b Combine like terms: 6b + -6b = 0 -7 + -6b + 2b2 = 0 Begin completing the square. Divide all terms by 2 the coefficient of the squared term: Divide each side by '2'. -3.5 + -3b + b2 = 0 Move the constant term to the right: Add '3.5' to each side of the equation. -3.5 + -3b + 3.5 + b2 = 0 + 3.5 Reorder the terms: -3.5 + 3.5 + -3b + b2 = 0 + 3.5 Combine like terms: -3.5 + 3.5 = 0.0 0.0 + -3b + b2 = 0 + 3.5 -3b + b2 = 0 + 3.5 Combine like terms: 0 + 3.5 = 3.5 -3b + b2 = 3.5 The b term is -3b. Take half its coefficient (-1.5). Square it (2.25) and add it to both sides. Add '2.25' to each side of the equation. -3b + 2.25 + b2 = 3.5 + 2.25 Reorder the terms: 2.25 + -3b + b2 = 3.5 + 2.25 Combine like terms: 3.5 + 2.25 = 5.75 2.25 + -3b + b2 = 5.75 Factor a perfect square on the left side: (b + -1.5)(b + -1.5) = 5.75 Calculate the square root of the right side: 2.397915762 Break this problem into two subproblems by setting (b + -1.5) equal to 2.397915762 and -2.397915762.Subproblem 1
b + -1.5 = 2.397915762 Simplifying b + -1.5 = 2.397915762 Reorder the terms: -1.5 + b = 2.397915762 Solving -1.5 + b = 2.397915762 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + b = 2.397915762 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + b = 2.397915762 + 1.5 b = 2.397915762 + 1.5 Combine like terms: 2.397915762 + 1.5 = 3.897915762 b = 3.897915762 Simplifying b = 3.897915762Subproblem 2
b + -1.5 = -2.397915762 Simplifying b + -1.5 = -2.397915762 Reorder the terms: -1.5 + b = -2.397915762 Solving -1.5 + b = -2.397915762 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '1.5' to each side of the equation. -1.5 + 1.5 + b = -2.397915762 + 1.5 Combine like terms: -1.5 + 1.5 = 0.0 0.0 + b = -2.397915762 + 1.5 b = -2.397915762 + 1.5 Combine like terms: -2.397915762 + 1.5 = -0.897915762 b = -0.897915762 Simplifying b = -0.897915762Solution
The solution to the problem is based on the solutions from the subproblems. b = {3.897915762, -0.897915762}
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