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Simplifying f(f + 6) = 23 Reorder the terms: f(6 + f) = 23 (6 * f + f * f) = 23 (6f + f2) = 23 Solving 6f + f2 = 23 Solving for variable 'f'. Reorder the terms: -23 + 6f + f2 = 23 + -23 Combine like terms: 23 + -23 = 0 -23 + 6f + f2 = 0 Begin completing the square. Move the constant term to the right: Add '23' to each side of the equation. -23 + 6f + 23 + f2 = 0 + 23 Reorder the terms: -23 + 23 + 6f + f2 = 0 + 23 Combine like terms: -23 + 23 = 0 0 + 6f + f2 = 0 + 23 6f + f2 = 0 + 23 Combine like terms: 0 + 23 = 23 6f + f2 = 23 The f term is 6f. Take half its coefficient (3). Square it (9) and add it to both sides. Add '9' to each side of the equation. 6f + 9 + f2 = 23 + 9 Reorder the terms: 9 + 6f + f2 = 23 + 9 Combine like terms: 23 + 9 = 32 9 + 6f + f2 = 32 Factor a perfect square on the left side: (f + 3)(f + 3) = 32 Calculate the square root of the right side: 5.656854249 Break this problem into two subproblems by setting (f + 3) equal to 5.656854249 and -5.656854249.Subproblem 1
f + 3 = 5.656854249 Simplifying f + 3 = 5.656854249 Reorder the terms: 3 + f = 5.656854249 Solving 3 + f = 5.656854249 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + f = 5.656854249 + -3 Combine like terms: 3 + -3 = 0 0 + f = 5.656854249 + -3 f = 5.656854249 + -3 Combine like terms: 5.656854249 + -3 = 2.656854249 f = 2.656854249 Simplifying f = 2.656854249Subproblem 2
f + 3 = -5.656854249 Simplifying f + 3 = -5.656854249 Reorder the terms: 3 + f = -5.656854249 Solving 3 + f = -5.656854249 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '-3' to each side of the equation. 3 + -3 + f = -5.656854249 + -3 Combine like terms: 3 + -3 = 0 0 + f = -5.656854249 + -3 f = -5.656854249 + -3 Combine like terms: -5.656854249 + -3 = -8.656854249 f = -8.656854249 Simplifying f = -8.656854249Solution
The solution to the problem is based on the solutions from the subproblems. f = {2.656854249, -8.656854249}
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