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Simplifying f2 + 28f = -5 Reorder the terms: 28f + f2 = -5 Solving 28f + f2 = -5 Solving for variable 'f'. Reorder the terms: 5 + 28f + f2 = -5 + 5 Combine like terms: -5 + 5 = 0 5 + 28f + f2 = 0 Begin completing the square. Move the constant term to the right: Add '-5' to each side of the equation. 5 + 28f + -5 + f2 = 0 + -5 Reorder the terms: 5 + -5 + 28f + f2 = 0 + -5 Combine like terms: 5 + -5 = 0 0 + 28f + f2 = 0 + -5 28f + f2 = 0 + -5 Combine like terms: 0 + -5 = -5 28f + f2 = -5 The f term is 28f. Take half its coefficient (14). Square it (196) and add it to both sides. Add '196' to each side of the equation. 28f + 196 + f2 = -5 + 196 Reorder the terms: 196 + 28f + f2 = -5 + 196 Combine like terms: -5 + 196 = 191 196 + 28f + f2 = 191 Factor a perfect square on the left side: (f + 14)(f + 14) = 191 Calculate the square root of the right side: 13.820274961 Break this problem into two subproblems by setting (f + 14) equal to 13.820274961 and -13.820274961.Subproblem 1
f + 14 = 13.820274961 Simplifying f + 14 = 13.820274961 Reorder the terms: 14 + f = 13.820274961 Solving 14 + f = 13.820274961 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '-14' to each side of the equation. 14 + -14 + f = 13.820274961 + -14 Combine like terms: 14 + -14 = 0 0 + f = 13.820274961 + -14 f = 13.820274961 + -14 Combine like terms: 13.820274961 + -14 = -0.179725039 f = -0.179725039 Simplifying f = -0.179725039Subproblem 2
f + 14 = -13.820274961 Simplifying f + 14 = -13.820274961 Reorder the terms: 14 + f = -13.820274961 Solving 14 + f = -13.820274961 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '-14' to each side of the equation. 14 + -14 + f = -13.820274961 + -14 Combine like terms: 14 + -14 = 0 0 + f = -13.820274961 + -14 f = -13.820274961 + -14 Combine like terms: -13.820274961 + -14 = -27.820274961 f = -27.820274961 Simplifying f = -27.820274961Solution
The solution to the problem is based on the solutions from the subproblems. f = {-0.179725039, -27.820274961}
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