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Simplifying f2 + -12f + -1 = 0 Reorder the terms: -1 + -12f + f2 = 0 Solving -1 + -12f + f2 = 0 Solving for variable 'f'. Begin completing the square. Move the constant term to the right: Add '1' to each side of the equation. -1 + -12f + 1 + f2 = 0 + 1 Reorder the terms: -1 + 1 + -12f + f2 = 0 + 1 Combine like terms: -1 + 1 = 0 0 + -12f + f2 = 0 + 1 -12f + f2 = 0 + 1 Combine like terms: 0 + 1 = 1 -12f + f2 = 1 The f term is -12f. Take half its coefficient (-6). Square it (36) and add it to both sides. Add '36' to each side of the equation. -12f + 36 + f2 = 1 + 36 Reorder the terms: 36 + -12f + f2 = 1 + 36 Combine like terms: 1 + 36 = 37 36 + -12f + f2 = 37 Factor a perfect square on the left side: (f + -6)(f + -6) = 37 Calculate the square root of the right side: 6.08276253 Break this problem into two subproblems by setting (f + -6) equal to 6.08276253 and -6.08276253.Subproblem 1
f + -6 = 6.08276253 Simplifying f + -6 = 6.08276253 Reorder the terms: -6 + f = 6.08276253 Solving -6 + f = 6.08276253 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '6' to each side of the equation. -6 + 6 + f = 6.08276253 + 6 Combine like terms: -6 + 6 = 0 0 + f = 6.08276253 + 6 f = 6.08276253 + 6 Combine like terms: 6.08276253 + 6 = 12.08276253 f = 12.08276253 Simplifying f = 12.08276253Subproblem 2
f + -6 = -6.08276253 Simplifying f + -6 = -6.08276253 Reorder the terms: -6 + f = -6.08276253 Solving -6 + f = -6.08276253 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '6' to each side of the equation. -6 + 6 + f = -6.08276253 + 6 Combine like terms: -6 + 6 = 0 0 + f = -6.08276253 + 6 f = -6.08276253 + 6 Combine like terms: -6.08276253 + 6 = -0.08276253 f = -0.08276253 Simplifying f = -0.08276253Solution
The solution to the problem is based on the solutions from the subproblems. f = {12.08276253, -0.08276253}
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