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Simplifying -2b(b + 7) + 5 = 0 Reorder the terms: -2b(7 + b) + 5 = 0 (7 * -2b + b * -2b) + 5 = 0 (-14b + -2b2) + 5 = 0 Reorder the terms: 5 + -14b + -2b2 = 0 Solving 5 + -14b + -2b2 = 0 Solving for variable 'b'. Begin completing the square. Divide all terms by -2 the coefficient of the squared term: Divide each side by '-2'. -2.5 + 7b + b2 = 0 Move the constant term to the right: Add '2.5' to each side of the equation. -2.5 + 7b + 2.5 + b2 = 0 + 2.5 Reorder the terms: -2.5 + 2.5 + 7b + b2 = 0 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + 7b + b2 = 0 + 2.5 7b + b2 = 0 + 2.5 Combine like terms: 0 + 2.5 = 2.5 7b + b2 = 2.5 The b term is 7b. Take half its coefficient (3.5). Square it (12.25) and add it to both sides. Add '12.25' to each side of the equation. 7b + 12.25 + b2 = 2.5 + 12.25 Reorder the terms: 12.25 + 7b + b2 = 2.5 + 12.25 Combine like terms: 2.5 + 12.25 = 14.75 12.25 + 7b + b2 = 14.75 Factor a perfect square on the left side: (b + 3.5)(b + 3.5) = 14.75 Calculate the square root of the right side: 3.840572874 Break this problem into two subproblems by setting (b + 3.5) equal to 3.840572874 and -3.840572874.Subproblem 1
b + 3.5 = 3.840572874 Simplifying b + 3.5 = 3.840572874 Reorder the terms: 3.5 + b = 3.840572874 Solving 3.5 + b = 3.840572874 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + b = 3.840572874 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + b = 3.840572874 + -3.5 b = 3.840572874 + -3.5 Combine like terms: 3.840572874 + -3.5 = 0.340572874 b = 0.340572874 Simplifying b = 0.340572874Subproblem 2
b + 3.5 = -3.840572874 Simplifying b + 3.5 = -3.840572874 Reorder the terms: 3.5 + b = -3.840572874 Solving 3.5 + b = -3.840572874 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-3.5' to each side of the equation. 3.5 + -3.5 + b = -3.840572874 + -3.5 Combine like terms: 3.5 + -3.5 = 0.0 0.0 + b = -3.840572874 + -3.5 b = -3.840572874 + -3.5 Combine like terms: -3.840572874 + -3.5 = -7.340572874 b = -7.340572874 Simplifying b = -7.340572874Solution
The solution to the problem is based on the solutions from the subproblems. b = {0.340572874, -7.340572874}
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