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Simplifying (10b2 + 5b + -2) + -1(2b2 + b + 1) = 0 Reorder the terms: (-2 + 5b + 10b2) + -1(2b2 + b + 1) = 0 Remove parenthesis around (-2 + 5b + 10b2) -2 + 5b + 10b2 + -1(2b2 + b + 1) = 0 Reorder the terms: -2 + 5b + 10b2 + -1(1 + b + 2b2) = 0 -2 + 5b + 10b2 + (1 * -1 + b * -1 + 2b2 * -1) = 0 -2 + 5b + 10b2 + (-1 + -1b + -2b2) = 0 Reorder the terms: -2 + -1 + 5b + -1b + 10b2 + -2b2 = 0 Combine like terms: -2 + -1 = -3 -3 + 5b + -1b + 10b2 + -2b2 = 0 Combine like terms: 5b + -1b = 4b -3 + 4b + 10b2 + -2b2 = 0 Combine like terms: 10b2 + -2b2 = 8b2 -3 + 4b + 8b2 = 0 Solving -3 + 4b + 8b2 = 0 Solving for variable 'b'. Begin completing the square. Divide all terms by 8 the coefficient of the squared term: Divide each side by '8'. -0.375 + 0.5b + b2 = 0 Move the constant term to the right: Add '0.375' to each side of the equation. -0.375 + 0.5b + 0.375 + b2 = 0 + 0.375 Reorder the terms: -0.375 + 0.375 + 0.5b + b2 = 0 + 0.375 Combine like terms: -0.375 + 0.375 = 0.000 0.000 + 0.5b + b2 = 0 + 0.375 0.5b + b2 = 0 + 0.375 Combine like terms: 0 + 0.375 = 0.375 0.5b + b2 = 0.375 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 = 0.375 + 0.0625 Reorder the terms: 0.0625 + 0.5b + b2 = 0.375 + 0.0625 Combine like terms: 0.375 + 0.0625 = 0.4375 0.0625 + 0.5b + b2 = 0.4375 Factor a perfect square on the left side: (b + 0.25)(b + 0.25) = 0.4375 Calculate the square root of the right side: 0.661437828 Break this problem into two subproblems by setting (b + 0.25) equal to 0.661437828 and -0.661437828.Subproblem 1
b + 0.25 = 0.661437828 Simplifying b + 0.25 = 0.661437828 Reorder the terms: 0.25 + b = 0.661437828 Solving 0.25 + b = 0.661437828 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 = 0.661437828 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + b = 0.661437828 + -0.25 b = 0.661437828 + -0.25 Combine like terms: 0.661437828 + -0.25 = 0.411437828 b = 0.411437828 Simplifying b = 0.411437828Subproblem 2
b + 0.25 = -0.661437828 Simplifying b + 0.25 = -0.661437828 Reorder the terms: 0.25 + b = -0.661437828 Solving 0.25 + b = -0.661437828 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 = -0.661437828 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + b = -0.661437828 + -0.25 b = -0.661437828 + -0.25 Combine like terms: -0.661437828 + -0.25 = -0.911437828 b = -0.911437828 Simplifying b = -0.911437828Solution
The solution to the problem is based on the solutions from the subproblems. b = {0.411437828, -0.911437828}
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