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Simplifying f2 + -4f + -23 = 0 Reorder the terms: -23 + -4f + f2 = 0 Solving -23 + -4f + f2 = 0 Solving for variable 'f'. Begin completing the square. Move the constant term to the right: Add '23' to each side of the equation. -23 + -4f + 23 + f2 = 0 + 23 Reorder the terms: -23 + 23 + -4f + f2 = 0 + 23 Combine like terms: -23 + 23 = 0 0 + -4f + f2 = 0 + 23 -4f + f2 = 0 + 23 Combine like terms: 0 + 23 = 23 -4f + f2 = 23 The f term is -4f. Take half its coefficient (-2). Square it (4) and add it to both sides. Add '4' to each side of the equation. -4f + 4 + f2 = 23 + 4 Reorder the terms: 4 + -4f + f2 = 23 + 4 Combine like terms: 23 + 4 = 27 4 + -4f + f2 = 27 Factor a perfect square on the left side: (f + -2)(f + -2) = 27 Calculate the square root of the right side: 5.196152423 Break this problem into two subproblems by setting (f + -2) equal to 5.196152423 and -5.196152423.Subproblem 1
f + -2 = 5.196152423 Simplifying f + -2 = 5.196152423 Reorder the terms: -2 + f = 5.196152423 Solving -2 + f = 5.196152423 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + f = 5.196152423 + 2 Combine like terms: -2 + 2 = 0 0 + f = 5.196152423 + 2 f = 5.196152423 + 2 Combine like terms: 5.196152423 + 2 = 7.196152423 f = 7.196152423 Simplifying f = 7.196152423Subproblem 2
f + -2 = -5.196152423 Simplifying f + -2 = -5.196152423 Reorder the terms: -2 + f = -5.196152423 Solving -2 + f = -5.196152423 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '2' to each side of the equation. -2 + 2 + f = -5.196152423 + 2 Combine like terms: -2 + 2 = 0 0 + f = -5.196152423 + 2 f = -5.196152423 + 2 Combine like terms: -5.196152423 + 2 = -3.196152423 f = -3.196152423 Simplifying f = -3.196152423Solution
The solution to the problem is based on the solutions from the subproblems. f = {7.196152423, -3.196152423}
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