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Simplifying f2 + -22f = -75 Reorder the terms: -22f + f2 = -75 Solving -22f + f2 = -75 Solving for variable 'f'. Reorder the terms: 75 + -22f + f2 = -75 + 75 Combine like terms: -75 + 75 = 0 75 + -22f + f2 = 0 Begin completing the square. Move the constant term to the right: Add '-75' to each side of the equation. 75 + -22f + -75 + f2 = 0 + -75 Reorder the terms: 75 + -75 + -22f + f2 = 0 + -75 Combine like terms: 75 + -75 = 0 0 + -22f + f2 = 0 + -75 -22f + f2 = 0 + -75 Combine like terms: 0 + -75 = -75 -22f + f2 = -75 The f term is -22f. Take half its coefficient (-11). Square it (121) and add it to both sides. Add '121' to each side of the equation. -22f + 121 + f2 = -75 + 121 Reorder the terms: 121 + -22f + f2 = -75 + 121 Combine like terms: -75 + 121 = 46 121 + -22f + f2 = 46 Factor a perfect square on the left side: (f + -11)(f + -11) = 46 Calculate the square root of the right side: 6.782329983 Break this problem into two subproblems by setting (f + -11) equal to 6.782329983 and -6.782329983.Subproblem 1
f + -11 = 6.782329983 Simplifying f + -11 = 6.782329983 Reorder the terms: -11 + f = 6.782329983 Solving -11 + f = 6.782329983 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '11' to each side of the equation. -11 + 11 + f = 6.782329983 + 11 Combine like terms: -11 + 11 = 0 0 + f = 6.782329983 + 11 f = 6.782329983 + 11 Combine like terms: 6.782329983 + 11 = 17.782329983 f = 17.782329983 Simplifying f = 17.782329983Subproblem 2
f + -11 = -6.782329983 Simplifying f + -11 = -6.782329983 Reorder the terms: -11 + f = -6.782329983 Solving -11 + f = -6.782329983 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '11' to each side of the equation. -11 + 11 + f = -6.782329983 + 11 Combine like terms: -11 + 11 = 0 0 + f = -6.782329983 + 11 f = -6.782329983 + 11 Combine like terms: -6.782329983 + 11 = 4.217670017 f = 4.217670017 Simplifying f = 4.217670017Solution
The solution to the problem is based on the solutions from the subproblems. f = {17.782329983, 4.217670017}
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