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Simplifying f2 + -16f = -5 Reorder the terms: -16f + f2 = -5 Solving -16f + f2 = -5 Solving for variable 'f'. Reorder the terms: 5 + -16f + f2 = -5 + 5 Combine like terms: -5 + 5 = 0 5 + -16f + f2 = 0 Begin completing the square. Move the constant term to the right: Add '-5' to each side of the equation. 5 + -16f + -5 + f2 = 0 + -5 Reorder the terms: 5 + -5 + -16f + f2 = 0 + -5 Combine like terms: 5 + -5 = 0 0 + -16f + f2 = 0 + -5 -16f + f2 = 0 + -5 Combine like terms: 0 + -5 = -5 -16f + f2 = -5 The f term is -16f. Take half its coefficient (-8). Square it (64) and add it to both sides. Add '64' to each side of the equation. -16f + 64 + f2 = -5 + 64 Reorder the terms: 64 + -16f + f2 = -5 + 64 Combine like terms: -5 + 64 = 59 64 + -16f + f2 = 59 Factor a perfect square on the left side: (f + -8)(f + -8) = 59 Calculate the square root of the right side: 7.681145748 Break this problem into two subproblems by setting (f + -8) equal to 7.681145748 and -7.681145748.Subproblem 1
f + -8 = 7.681145748 Simplifying f + -8 = 7.681145748 Reorder the terms: -8 + f = 7.681145748 Solving -8 + f = 7.681145748 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '8' to each side of the equation. -8 + 8 + f = 7.681145748 + 8 Combine like terms: -8 + 8 = 0 0 + f = 7.681145748 + 8 f = 7.681145748 + 8 Combine like terms: 7.681145748 + 8 = 15.681145748 f = 15.681145748 Simplifying f = 15.681145748Subproblem 2
f + -8 = -7.681145748 Simplifying f + -8 = -7.681145748 Reorder the terms: -8 + f = -7.681145748 Solving -8 + f = -7.681145748 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '8' to each side of the equation. -8 + 8 + f = -7.681145748 + 8 Combine like terms: -8 + 8 = 0 0 + f = -7.681145748 + 8 f = -7.681145748 + 8 Combine like terms: -7.681145748 + 8 = 0.318854252 f = 0.318854252 Simplifying f = 0.318854252Solution
The solution to the problem is based on the solutions from the subproblems. f = {15.681145748, 0.318854252}
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