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Simplifying 56 = 0.5b2 + 6b Reorder the terms: 56 = 6b + 0.5b2 Solving 56 = 6b + 0.5b2 Solving for variable 'b'. Reorder the terms: 56 + -6b + -0.5b2 = 6b + -6b + 0.5b2 + -0.5b2 Combine like terms: 6b + -6b = 0 56 + -6b + -0.5b2 = 0 + 0.5b2 + -0.5b2 56 + -6b + -0.5b2 = 0.5b2 + -0.5b2 Combine like terms: 0.5b2 + -0.5b2 = 0.0 56 + -6b + -0.5b2 = 0.0 Begin completing the square. Divide all terms by -0.5 the coefficient of the squared term: Divide each side by '-0.5'. -112 + 12b + b2 = 0 Move the constant term to the right: Add '112' to each side of the equation. -112 + 12b + 112 + b2 = 0 + 112 Reorder the terms: -112 + 112 + 12b + b2 = 0 + 112 Combine like terms: -112 + 112 = 0 0 + 12b + b2 = 0 + 112 12b + b2 = 0 + 112 Combine like terms: 0 + 112 = 112 12b + b2 = 112 The b term is 12b. Take half its coefficient (6). Square it (36) and add it to both sides. Add '36' to each side of the equation. 12b + 36 + b2 = 112 + 36 Reorder the terms: 36 + 12b + b2 = 112 + 36 Combine like terms: 112 + 36 = 148 36 + 12b + b2 = 148 Factor a perfect square on the left side: (b + 6)(b + 6) = 148 Calculate the square root of the right side: 12.165525061 Break this problem into two subproblems by setting (b + 6) equal to 12.165525061 and -12.165525061.Subproblem 1
b + 6 = 12.165525061 Simplifying b + 6 = 12.165525061 Reorder the terms: 6 + b = 12.165525061 Solving 6 + b = 12.165525061 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-6' to each side of the equation. 6 + -6 + b = 12.165525061 + -6 Combine like terms: 6 + -6 = 0 0 + b = 12.165525061 + -6 b = 12.165525061 + -6 Combine like terms: 12.165525061 + -6 = 6.165525061 b = 6.165525061 Simplifying b = 6.165525061Subproblem 2
b + 6 = -12.165525061 Simplifying b + 6 = -12.165525061 Reorder the terms: 6 + b = -12.165525061 Solving 6 + b = -12.165525061 Solving for variable 'b'. Move all terms containing b to the left, all other terms to the right. Add '-6' to each side of the equation. 6 + -6 + b = -12.165525061 + -6 Combine like terms: 6 + -6 = 0 0 + b = -12.165525061 + -6 b = -12.165525061 + -6 Combine like terms: -12.165525061 + -6 = -18.165525061 b = -18.165525061 Simplifying b = -18.165525061Solution
The solution to the problem is based on the solutions from the subproblems. b = {6.165525061, -18.165525061}
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