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Simplifying b2 + 0.5b = 3 Reorder the terms: 0.5b + b2 = 3 Solving 0.5b + b2 = 3 Solving for variable 'b'. Reorder the terms: -3 + 0.5b + b2 = 3 + -3 Combine like terms: 3 + -3 = 0 -3 + 0.5b + b2 = 0 Begin completing the square. Move the constant term to the right: Add '3' to each side of the equation. -3 + 0.5b + 3 + b2 = 0 + 3 Reorder the terms: -3 + 3 + 0.5b + b2 = 0 + 3 Combine like terms: -3 + 3 = 0 0 + 0.5b + b2 = 0 + 3 0.5b + b2 = 0 + 3 Combine like terms: 0 + 3 = 3 0.5b + b2 = 3 The b term is 0.5b. Take half its coefficient (0.25). Square it (0.0625) and add it to both sides. Add '0.0625' to each side of the equation. 0.5b + 0.0625 + b2 = 3 + 0.0625 Reorder the terms: 0.0625 + 0.5b + b2 = 3 + 0.0625 Combine like terms: 3 + 0.0625 = 3.0625 0.0625 + 0.5b + b2 = 3.0625 Factor a perfect square on the left side: (b + 0.25)(b + 0.25) = 3.0625 Calculate the square root of the right side: 1.75 Break this problem into two subproblems by setting (b + 0.25) equal to 1.75 and -1.75.Subproblem 1
b + 0.25 = 1.75 Simplifying b + 0.25 = 1.75 Reorder the terms: 0.25 + b = 1.75 Solving 0.25 + b = 1.75 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + b = 1.75 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + b = 1.75 + -0.25 b = 1.75 + -0.25 Combine like terms: 1.75 + -0.25 = 1.5 b = 1.5 Simplifying b = 1.5Subproblem 2
b + 0.25 = -1.75 Simplifying b + 0.25 = -1.75 Reorder the terms: 0.25 + b = -1.75 Solving 0.25 + b = -1.75 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-0.25' to each side of the equation. 0.25 + -0.25 + b = -1.75 + -0.25 Combine like terms: 0.25 + -0.25 = 0.00 0.00 + b = -1.75 + -0.25 b = -1.75 + -0.25 Combine like terms: -1.75 + -0.25 = -2 b = -2 Simplifying b = -2Solution
The solution to the problem is based on the solutions from the subproblems. b = {1.5, -2}
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