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Simplifying -2(-2b + 5) * 8b = 14 Reorder the terms: -2(5 + -2b) * 8b = 14 Reorder the terms for easier multiplication: -2 * 8b(5 + -2b) = 14 Multiply -2 * 8 -16b(5 + -2b) = 14 (5 * -16b + -2b * -16b) = 14 (-80b + 32b2) = 14 Solving -80b + 32b2 = 14 Solving for variable 'b'. Reorder the terms: -14 + -80b + 32b2 = 14 + -14 Combine like terms: 14 + -14 = 0 -14 + -80b + 32b2 = 0 Factor out the Greatest Common Factor (GCF), '2'. 2(-7 + -40b + 16b2) = 0 Ignore the factor 2.Subproblem 1
Set the factor '(-7 + -40b + 16b2)' equal to zero and attempt to solve: Simplifying -7 + -40b + 16b2 = 0 Solving -7 + -40b + 16b2 = 0 Begin completing the square. Divide all terms by 16 the coefficient of the squared term: Divide each side by '16'. -0.4375 + -2.5b + b2 = 0 Move the constant term to the right: Add '0.4375' to each side of the equation. -0.4375 + -2.5b + 0.4375 + b2 = 0 + 0.4375 Reorder the terms: -0.4375 + 0.4375 + -2.5b + b2 = 0 + 0.4375 Combine like terms: -0.4375 + 0.4375 = 0.0000 0.0000 + -2.5b + b2 = 0 + 0.4375 -2.5b + b2 = 0 + 0.4375 Combine like terms: 0 + 0.4375 = 0.4375 -2.5b + b2 = 0.4375 The b term is -2.5b. Take half its coefficient (-1.25). Square it (1.5625) and add it to both sides. Add '1.5625' to each side of the equation. -2.5b + 1.5625 + b2 = 0.4375 + 1.5625 Reorder the terms: 1.5625 + -2.5b + b2 = 0.4375 + 1.5625 Combine like terms: 0.4375 + 1.5625 = 2 1.5625 + -2.5b + b2 = 2 Factor a perfect square on the left side: (b + -1.25)(b + -1.25) = 2 Calculate the square root of the right side: 1.414213562 Break this problem into two subproblems by setting (b + -1.25) equal to 1.414213562 and -1.414213562.Subproblem 1
b + -1.25 = 1.414213562 Simplifying b + -1.25 = 1.414213562 Reorder the terms: -1.25 + b = 1.414213562 Solving -1.25 + b = 1.414213562 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + b = 1.414213562 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + b = 1.414213562 + 1.25 b = 1.414213562 + 1.25 Combine like terms: 1.414213562 + 1.25 = 2.664213562 b = 2.664213562 Simplifying b = 2.664213562Subproblem 2
b + -1.25 = -1.414213562 Simplifying b + -1.25 = -1.414213562 Reorder the terms: -1.25 + b = -1.414213562 Solving -1.25 + b = -1.414213562 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '1.25' to each side of the equation. -1.25 + 1.25 + b = -1.414213562 + 1.25 Combine like terms: -1.25 + 1.25 = 0.00 0.00 + b = -1.414213562 + 1.25 b = -1.414213562 + 1.25 Combine like terms: -1.414213562 + 1.25 = -0.164213562 b = -0.164213562 Simplifying b = -0.164213562Solution
The solution to the problem is based on the solutions from the subproblems. b = {2.664213562, -0.164213562}Solution
b = {2.664213562, -0.164213562}
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