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Simplifying 5b2 + 13b + 8 = b2 Reorder the terms: 8 + 13b + 5b2 = b2 Solving 8 + 13b + 5b2 = b2 Solving for variable 'b'. Combine like terms: 5b2 + -1b2 = 4b2 8 + 13b + 4b2 = b2 + -1b2 Combine like terms: b2 + -1b2 = 0 8 + 13b + 4b2 = 0 Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. 2 + 3.25b + b2 = 0 Move the constant term to the right: Add '-2' to each side of the equation. 2 + 3.25b + -2 + b2 = 0 + -2 Reorder the terms: 2 + -2 + 3.25b + b2 = 0 + -2 Combine like terms: 2 + -2 = 0 0 + 3.25b + b2 = 0 + -2 3.25b + b2 = 0 + -2 Combine like terms: 0 + -2 = -2 3.25b + b2 = -2 The b term is 3.25b. Take half its coefficient (1.625). Square it (2.640625) and add it to both sides. Add '2.640625' to each side of the equation. 3.25b + 2.640625 + b2 = -2 + 2.640625 Reorder the terms: 2.640625 + 3.25b + b2 = -2 + 2.640625 Combine like terms: -2 + 2.640625 = 0.640625 2.640625 + 3.25b + b2 = 0.640625 Factor a perfect square on the left side: (b + 1.625)(b + 1.625) = 0.640625 Calculate the square root of the right side: 0.80039053 Break this problem into two subproblems by setting (b + 1.625) equal to 0.80039053 and -0.80039053.Subproblem 1
b + 1.625 = 0.80039053 Simplifying b + 1.625 = 0.80039053 Reorder the terms: 1.625 + b = 0.80039053 Solving 1.625 + b = 0.80039053 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-1.625' to each side of the equation. 1.625 + -1.625 + b = 0.80039053 + -1.625 Combine like terms: 1.625 + -1.625 = 0.000 0.000 + b = 0.80039053 + -1.625 b = 0.80039053 + -1.625 Combine like terms: 0.80039053 + -1.625 = -0.82460947 b = -0.82460947 Simplifying b = -0.82460947Subproblem 2
b + 1.625 = -0.80039053 Simplifying b + 1.625 = -0.80039053 Reorder the terms: 1.625 + b = -0.80039053 Solving 1.625 + b = -0.80039053 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-1.625' to each side of the equation. 1.625 + -1.625 + b = -0.80039053 + -1.625 Combine like terms: 1.625 + -1.625 = 0.000 0.000 + b = -0.80039053 + -1.625 b = -0.80039053 + -1.625 Combine like terms: -0.80039053 + -1.625 = -2.42539053 b = -2.42539053 Simplifying b = -2.42539053Solution
The solution to the problem is based on the solutions from the subproblems. b = {-0.82460947, -2.42539053}
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