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Simplifying 4b2 + -7 = -4b Reorder the terms: -7 + 4b2 = -4b Solving -7 + 4b2 = -4b Solving for variable 'b'. Reorder the terms: -7 + 4b + 4b2 = -4b + 4b Combine like terms: -4b + 4b = 0 -7 + 4b + 4b2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -1.75 + b + b2 = 0 Move the constant term to the right: Add '1.75' to each side of the equation. -1.75 + b + 1.75 + b2 = 0 + 1.75 Reorder the terms: -1.75 + 1.75 + b + b2 = 0 + 1.75 Combine like terms: -1.75 + 1.75 = 0.00 0.00 + b + b2 = 0 + 1.75 b + b2 = 0 + 1.75 Combine like terms: 0 + 1.75 = 1.75 b + b2 = 1.75 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 = 1.75 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + b2 = 1.75 + 0.25 Combine like terms: 1.75 + 0.25 = 2 1.25 + b2 = 2 Factor a perfect square on the left side: (b + 0.5)(b + 0.5) = 2 Calculate the square root of the right side: 1.414213562 Break this problem into two subproblems by setting (b + 0.5) equal to 1.414213562 and -1.414213562.Subproblem 1
b + 0.5 = 1.414213562 Simplifying b + 0.5 = 1.414213562 Reorder the terms: 0.5 + b = 1.414213562 Solving 0.5 + b = 1.414213562 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 = 1.414213562 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + b = 1.414213562 + -0.5 b = 1.414213562 + -0.5 Combine like terms: 1.414213562 + -0.5 = 0.914213562 b = 0.914213562 Simplifying b = 0.914213562Subproblem 2
b + 0.5 = -1.414213562 Simplifying b + 0.5 = -1.414213562 Reorder the terms: 0.5 + b = -1.414213562 Solving 0.5 + b = -1.414213562 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 = -1.414213562 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + b = -1.414213562 + -0.5 b = -1.414213562 + -0.5 Combine like terms: -1.414213562 + -0.5 = -1.914213562 b = -1.914213562 Simplifying b = -1.914213562Solution
The solution to the problem is based on the solutions from the subproblems. b = {0.914213562, -1.914213562}
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