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Simplifying 42 = b2 + -5b Reorder the terms: 42 = -5b + b2 Solving 42 = -5b + b2 Solving for variable 'b'. Reorder the terms: 42 + 5b + -1b2 = -5b + 5b + b2 + -1b2 Combine like terms: -5b + 5b = 0 42 + 5b + -1b2 = 0 + b2 + -1b2 42 + 5b + -1b2 = b2 + -1b2 Combine like terms: b2 + -1b2 = 0 42 + 5b + -1b2 = 0 Begin completing the square. Divide all terms by -1 the coefficient of the squared term: Divide each side by '-1'. -42 + -5b + b2 = 0 Move the constant term to the right: Add '42' to each side of the equation. -42 + -5b + 42 + b2 = 0 + 42 Reorder the terms: -42 + 42 + -5b + b2 = 0 + 42 Combine like terms: -42 + 42 = 0 0 + -5b + b2 = 0 + 42 -5b + b2 = 0 + 42 Combine like terms: 0 + 42 = 42 -5b + b2 = 42 The b term is -5b. Take half its coefficient (-2.5). Square it (6.25) and add it to both sides. Add '6.25' to each side of the equation. -5b + 6.25 + b2 = 42 + 6.25 Reorder the terms: 6.25 + -5b + b2 = 42 + 6.25 Combine like terms: 42 + 6.25 = 48.25 6.25 + -5b + b2 = 48.25 Factor a perfect square on the left side: (b + -2.5)(b + -2.5) = 48.25 Calculate the square root of the right side: 6.946221995 Break this problem into two subproblems by setting (b + -2.5) equal to 6.946221995 and -6.946221995.Subproblem 1
b + -2.5 = 6.946221995 Simplifying b + -2.5 = 6.946221995 Reorder the terms: -2.5 + b = 6.946221995 Solving -2.5 + b = 6.946221995 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + b = 6.946221995 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + b = 6.946221995 + 2.5 b = 6.946221995 + 2.5 Combine like terms: 6.946221995 + 2.5 = 9.446221995 b = 9.446221995 Simplifying b = 9.446221995Subproblem 2
b + -2.5 = -6.946221995 Simplifying b + -2.5 = -6.946221995 Reorder the terms: -2.5 + b = -6.946221995 Solving -2.5 + b = -6.946221995 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '2.5' to each side of the equation. -2.5 + 2.5 + b = -6.946221995 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + b = -6.946221995 + 2.5 b = -6.946221995 + 2.5 Combine like terms: -6.946221995 + 2.5 = -4.446221995 b = -4.446221995 Simplifying b = -4.446221995Solution
The solution to the problem is based on the solutions from the subproblems. b = {9.446221995, -4.446221995}
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