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Simplifying f2 + 18f + -31 = 0 Reorder the terms: -31 + 18f + f2 = 0 Solving -31 + 18f + f2 = 0 Solving for variable 'f'. Begin completing the square. Move the constant term to the right: Add '31' to each side of the equation. -31 + 18f + 31 + f2 = 0 + 31 Reorder the terms: -31 + 31 + 18f + f2 = 0 + 31 Combine like terms: -31 + 31 = 0 0 + 18f + f2 = 0 + 31 18f + f2 = 0 + 31 Combine like terms: 0 + 31 = 31 18f + f2 = 31 The f term is 18f. Take half its coefficient (9). Square it (81) and add it to both sides. Add '81' to each side of the equation. 18f + 81 + f2 = 31 + 81 Reorder the terms: 81 + 18f + f2 = 31 + 81 Combine like terms: 31 + 81 = 112 81 + 18f + f2 = 112 Factor a perfect square on the left side: (f + 9)(f + 9) = 112 Calculate the square root of the right side: 10.583005244 Break this problem into two subproblems by setting (f + 9) equal to 10.583005244 and -10.583005244.Subproblem 1
f + 9 = 10.583005244 Simplifying f + 9 = 10.583005244 Reorder the terms: 9 + f = 10.583005244 Solving 9 + f = 10.583005244 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '-9' to each side of the equation. 9 + -9 + f = 10.583005244 + -9 Combine like terms: 9 + -9 = 0 0 + f = 10.583005244 + -9 f = 10.583005244 + -9 Combine like terms: 10.583005244 + -9 = 1.583005244 f = 1.583005244 Simplifying f = 1.583005244Subproblem 2
f + 9 = -10.583005244 Simplifying f + 9 = -10.583005244 Reorder the terms: 9 + f = -10.583005244 Solving 9 + f = -10.583005244 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '-9' to each side of the equation. 9 + -9 + f = -10.583005244 + -9 Combine like terms: 9 + -9 = 0 0 + f = -10.583005244 + -9 f = -10.583005244 + -9 Combine like terms: -10.583005244 + -9 = -19.583005244 f = -19.583005244 Simplifying f = -19.583005244Solution
The solution to the problem is based on the solutions from the subproblems. f = {1.583005244, -19.583005244}
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