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Simplifying 0.5(b2 + 2b) = 24 Reorder the terms: 0.5(2b + b2) = 24 (2b * 0.5 + b2 * 0.5) = 24 (1b + 0.5b2) = 24 Solving 1b + 0.5b2 = 24 Solving for variable 'b'. Reorder the terms: -24 + 1b + 0.5b2 = 24 + -24 Combine like terms: 24 + -24 = 0 -24 + 1b + 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'. -48 + 2b + b2 = 0 Move the constant term to the right: Add '48' to each side of the equation. -48 + 2b + 48 + b2 = 0 + 48 Reorder the terms: -48 + 48 + 2b + b2 = 0 + 48 Combine like terms: -48 + 48 = 0 0 + 2b + b2 = 0 + 48 2b + b2 = 0 + 48 Combine like terms: 0 + 48 = 48 2b + b2 = 48 The b term is 2b. Take half its coefficient (1). Square it (1) and add it to both sides. Add '1' to each side of the equation. 2b + 1 + b2 = 48 + 1 Reorder the terms: 1 + 2b + b2 = 48 + 1 Combine like terms: 48 + 1 = 49 1 + 2b + b2 = 49 Factor a perfect square on the left side: (b + 1)(b + 1) = 49 Calculate the square root of the right side: 7 Break this problem into two subproblems by setting (b + 1) equal to 7 and -7.Subproblem 1
b + 1 = 7 Simplifying b + 1 = 7 Reorder the terms: 1 + b = 7 Solving 1 + b = 7 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + b = 7 + -1 Combine like terms: 1 + -1 = 0 0 + b = 7 + -1 b = 7 + -1 Combine like terms: 7 + -1 = 6 b = 6 Simplifying b = 6Subproblem 2
b + 1 = -7 Simplifying b + 1 = -7 Reorder the terms: 1 + b = -7 Solving 1 + b = -7 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + b = -7 + -1 Combine like terms: 1 + -1 = 0 0 + b = -7 + -1 b = -7 + -1 Combine like terms: -7 + -1 = -8 b = -8 Simplifying b = -8Solution
The solution to the problem is based on the solutions from the subproblems. b = {6, -8}
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