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Simplifying 0.5(b)(b + -5) = 42 Reorder the terms: 0.5b(-5 + b) = 42 (-5 * 0.5b + b * 0.5b) = 42 (-2.5b + 0.5b2) = 42 Solving -2.5b + 0.5b2 = 42 Solving for variable 'b'. Reorder the terms: -42 + -2.5b + 0.5b2 = 42 + -42 Combine like terms: 42 + -42 = 0 -42 + -2.5b + 0.5b2 = 0 Begin completing the square. Divide all terms by 0.5 the coefficient of the squared term: Divide each side by '0.5'. -84 + -5b + b2 = 0 Move the constant term to the right: Add '84' to each side of the equation. -84 + -5b + 84 + b2 = 0 + 84 Reorder the terms: -84 + 84 + -5b + b2 = 0 + 84 Combine like terms: -84 + 84 = 0 0 + -5b + b2 = 0 + 84 -5b + b2 = 0 + 84 Combine like terms: 0 + 84 = 84 -5b + b2 = 84 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 = 84 + 6.25 Reorder the terms: 6.25 + -5b + b2 = 84 + 6.25 Combine like terms: 84 + 6.25 = 90.25 6.25 + -5b + b2 = 90.25 Factor a perfect square on the left side: (b + -2.5)(b + -2.5) = 90.25 Calculate the square root of the right side: 9.5 Break this problem into two subproblems by setting (b + -2.5) equal to 9.5 and -9.5.Subproblem 1
b + -2.5 = 9.5 Simplifying b + -2.5 = 9.5 Reorder the terms: -2.5 + b = 9.5 Solving -2.5 + b = 9.5 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 = 9.5 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + b = 9.5 + 2.5 b = 9.5 + 2.5 Combine like terms: 9.5 + 2.5 = 12 b = 12 Simplifying b = 12Subproblem 2
b + -2.5 = -9.5 Simplifying b + -2.5 = -9.5 Reorder the terms: -2.5 + b = -9.5 Solving -2.5 + b = -9.5 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 = -9.5 + 2.5 Combine like terms: -2.5 + 2.5 = 0.0 0.0 + b = -9.5 + 2.5 b = -9.5 + 2.5 Combine like terms: -9.5 + 2.5 = -7 b = -7 Simplifying b = -7Solution
The solution to the problem is based on the solutions from the subproblems. b = {12, -7}
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