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Simplifying 7 + b(5 + -5b) + -10b = 0 7 + (5 * b + -5b * b) + -10b = 0 7 + (5b + -5b2) + -10b = 0 Reorder the terms: 7 + 5b + -10b + -5b2 = 0 Combine like terms: 5b + -10b = -5b 7 + -5b + -5b2 = 0 Solving 7 + -5b + -5b2 = 0 Solving for variable 'b'. Begin completing the square. Divide all terms by -5 the coefficient of the squared term: Divide each side by '-5'. -1.4 + b + b2 = 0 Move the constant term to the right: Add '1.4' to each side of the equation. -1.4 + b + 1.4 + b2 = 0 + 1.4 Reorder the terms: -1.4 + 1.4 + b + b2 = 0 + 1.4 Combine like terms: -1.4 + 1.4 = 0.0 0.0 + b + b2 = 0 + 1.4 b + b2 = 0 + 1.4 Combine like terms: 0 + 1.4 = 1.4 b + b2 = 1.4 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 = 1.4 + 0.25 Combine like terms: + 0.25 = 1.25 1.25 + b2 = 1.4 + 0.25 Combine like terms: 1.4 + 0.25 = 1.65 1.25 + b2 = 1.65 Factor a perfect square on the left side: (b + 0.5)(b + 0.5) = 1.65 Calculate the square root of the right side: 1.284523258 Break this problem into two subproblems by setting (b + 0.5) equal to 1.284523258 and -1.284523258.Subproblem 1
b + 0.5 = 1.284523258 Simplifying b + 0.5 = 1.284523258 Reorder the terms: 0.5 + b = 1.284523258 Solving 0.5 + b = 1.284523258 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 = 1.284523258 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + b = 1.284523258 + -0.5 b = 1.284523258 + -0.5 Combine like terms: 1.284523258 + -0.5 = 0.784523258 b = 0.784523258 Simplifying b = 0.784523258Subproblem 2
b + 0.5 = -1.284523258 Simplifying b + 0.5 = -1.284523258 Reorder the terms: 0.5 + b = -1.284523258 Solving 0.5 + b = -1.284523258 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 = -1.284523258 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + b = -1.284523258 + -0.5 b = -1.284523258 + -0.5 Combine like terms: -1.284523258 + -0.5 = -1.784523258 b = -1.784523258 Simplifying b = -1.784523258Solution
The solution to the problem is based on the solutions from the subproblems. b = {0.784523258, -1.784523258}
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