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Simplifying 8f + f(f + -6) = 57 + -1(17 + -1f) Reorder the terms: 8f + f(-6 + f) = 57 + -1(17 + -1f) 8f + (-6 * f + f * f) = 57 + -1(17 + -1f) 8f + (-6f + f2) = 57 + -1(17 + -1f) Combine like terms: 8f + -6f = 2f 2f + f2 = 57 + -1(17 + -1f) 2f + f2 = 57 + (17 * -1 + -1f * -1) 2f + f2 = 57 + (-17 + 1f) Combine like terms: 57 + -17 = 40 2f + f2 = 40 + 1f Solving 2f + f2 = 40 + 1f Solving for variable 'f'. Reorder the terms: -40 + 2f + -1f + f2 = 40 + 1f + -40 + -1f Combine like terms: 2f + -1f = 1f -40 + 1f + f2 = 40 + 1f + -40 + -1f Reorder the terms: -40 + 1f + f2 = 40 + -40 + 1f + -1f Combine like terms: 40 + -40 = 0 -40 + 1f + f2 = 0 + 1f + -1f -40 + 1f + f2 = 1f + -1f Combine like terms: 1f + -1f = 0 -40 + 1f + f2 = 0 Begin completing the square. Move the constant term to the right: Add '40' to each side of the equation. -40 + 1f + 40 + f2 = 0 + 40 Reorder the terms: -40 + 40 + 1f + f2 = 0 + 40 Combine like terms: -40 + 40 = 0 0 + 1f + f2 = 0 + 40 1f + f2 = 0 + 40 Combine like terms: 0 + 40 = 40 1f + f2 = 40 The f term is 1f. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. f + 0.25 + f2 = 40 + 0.25 Reorder the terms: 0.25 + f + f2 = 40 + 0.25 Combine like terms: 40 + 0.25 = 40.25 0.25 + f + f2 = 40.25 Factor a perfect square on the left side: (f + 0.5)(f + 0.5) = 40.25 Calculate the square root of the right side: 6.34428877 Break this problem into two subproblems by setting (f + 0.5) equal to 6.34428877 and -6.34428877.Subproblem 1
f + 0.5 = 6.34428877 Simplifying f + 0.5 = 6.34428877 Reorder the terms: 0.5 + f = 6.34428877 Solving 0.5 + f = 6.34428877 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + f = 6.34428877 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + f = 6.34428877 + -0.5 f = 6.34428877 + -0.5 Combine like terms: 6.34428877 + -0.5 = 5.84428877 f = 5.84428877 Simplifying f = 5.84428877Subproblem 2
f + 0.5 = -6.34428877 Simplifying f + 0.5 = -6.34428877 Reorder the terms: 0.5 + f = -6.34428877 Solving 0.5 + f = -6.34428877 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + f = -6.34428877 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + f = -6.34428877 + -0.5 f = -6.34428877 + -0.5 Combine like terms: -6.34428877 + -0.5 = -6.84428877 f = -6.84428877 Simplifying f = -6.84428877Solution
The solution to the problem is based on the solutions from the subproblems. f = {5.84428877, -6.84428877}
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