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Simplifying f(4 + f) + 4f = 860 (4 * f + f * f) + 4f = 860 (4f + f2) + 4f = 860 Reorder the terms: 4f + 4f + f2 = 860 Combine like terms: 4f + 4f = 8f 8f + f2 = 860 Solving 8f + f2 = 860 Solving for variable 'f'. Reorder the terms: -860 + 8f + f2 = 860 + -860 Combine like terms: 860 + -860 = 0 -860 + 8f + f2 = 0 Begin completing the square. Move the constant term to the right: Add '860' to each side of the equation. -860 + 8f + 860 + f2 = 0 + 860 Reorder the terms: -860 + 860 + 8f + f2 = 0 + 860 Combine like terms: -860 + 860 = 0 0 + 8f + f2 = 0 + 860 8f + f2 = 0 + 860 Combine like terms: 0 + 860 = 860 8f + f2 = 860 The f term is 8f. Take half its coefficient (4). Square it (16) and add it to both sides. Add '16' to each side of the equation. 8f + 16 + f2 = 860 + 16 Reorder the terms: 16 + 8f + f2 = 860 + 16 Combine like terms: 860 + 16 = 876 16 + 8f + f2 = 876 Factor a perfect square on the left side: (f + 4)(f + 4) = 876 Calculate the square root of the right side: 29.597297174 Break this problem into two subproblems by setting (f + 4) equal to 29.597297174 and -29.597297174.Subproblem 1
f + 4 = 29.597297174 Simplifying f + 4 = 29.597297174 Reorder the terms: 4 + f = 29.597297174 Solving 4 + f = 29.597297174 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '-4' to each side of the equation. 4 + -4 + f = 29.597297174 + -4 Combine like terms: 4 + -4 = 0 0 + f = 29.597297174 + -4 f = 29.597297174 + -4 Combine like terms: 29.597297174 + -4 = 25.597297174 f = 25.597297174 Simplifying f = 25.597297174Subproblem 2
f + 4 = -29.597297174 Simplifying f + 4 = -29.597297174 Reorder the terms: 4 + f = -29.597297174 Solving 4 + f = -29.597297174 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '-4' to each side of the equation. 4 + -4 + f = -29.597297174 + -4 Combine like terms: 4 + -4 = 0 0 + f = -29.597297174 + -4 f = -29.597297174 + -4 Combine like terms: -29.597297174 + -4 = -33.597297174 f = -33.597297174 Simplifying f = -33.597297174Solution
The solution to the problem is based on the solutions from the subproblems. f = {25.597297174, -33.597297174}
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