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Simplifying f(f + 6) = 30 Reorder the terms: f(6 + f) = 30 (6 * f + f * f) = 30 (6f + f2) = 30 Solving 6f + f2 = 30 Solving for variable 'f'. Reorder the terms: -30 + 6f + f2 = 30 + -30 Combine like terms: 30 + -30 = 0 -30 + 6f + f2 = 0 Begin completing the square. Move the constant term to the right: Add '30' to each side of the equation. -30 + 6f + 30 + f2 = 0 + 30 Reorder the terms: -30 + 30 + 6f + f2 = 0 + 30 Combine like terms: -30 + 30 = 0 0 + 6f + f2 = 0 + 30 6f + f2 = 0 + 30 Combine like terms: 0 + 30 = 30 6f + f2 = 30 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 = 30 + 9 Reorder the terms: 9 + 6f + f2 = 30 + 9 Combine like terms: 30 + 9 = 39 9 + 6f + f2 = 39 Factor a perfect square on the left side: (f + 3)(f + 3) = 39 Calculate the square root of the right side: 6.244997998 Break this problem into two subproblems by setting (f + 3) equal to 6.244997998 and -6.244997998.Subproblem 1
f + 3 = 6.244997998 Simplifying f + 3 = 6.244997998 Reorder the terms: 3 + f = 6.244997998 Solving 3 + f = 6.244997998 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 = 6.244997998 + -3 Combine like terms: 3 + -3 = 0 0 + f = 6.244997998 + -3 f = 6.244997998 + -3 Combine like terms: 6.244997998 + -3 = 3.244997998 f = 3.244997998 Simplifying f = 3.244997998Subproblem 2
f + 3 = -6.244997998 Simplifying f + 3 = -6.244997998 Reorder the terms: 3 + f = -6.244997998 Solving 3 + f = -6.244997998 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 = -6.244997998 + -3 Combine like terms: 3 + -3 = 0 0 + f = -6.244997998 + -3 f = -6.244997998 + -3 Combine like terms: -6.244997998 + -3 = -9.244997998 f = -9.244997998 Simplifying f = -9.244997998Solution
The solution to the problem is based on the solutions from the subproblems. f = {3.244997998, -9.244997998}
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