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Simplifying f2 + 10f = -1 Reorder the terms: 10f + f2 = -1 Solving 10f + f2 = -1 Solving for variable 'f'. Reorder the terms: 1 + 10f + f2 = -1 + 1 Combine like terms: -1 + 1 = 0 1 + 10f + f2 = 0 Begin completing the square. Move the constant term to the right: Add '-1' to each side of the equation. 1 + 10f + -1 + f2 = 0 + -1 Reorder the terms: 1 + -1 + 10f + f2 = 0 + -1 Combine like terms: 1 + -1 = 0 0 + 10f + f2 = 0 + -1 10f + f2 = 0 + -1 Combine like terms: 0 + -1 = -1 10f + f2 = -1 The f term is 10f. Take half its coefficient (5). Square it (25) and add it to both sides. Add '25' to each side of the equation. 10f + 25 + f2 = -1 + 25 Reorder the terms: 25 + 10f + f2 = -1 + 25 Combine like terms: -1 + 25 = 24 25 + 10f + f2 = 24 Factor a perfect square on the left side: (f + 5)(f + 5) = 24 Calculate the square root of the right side: 4.898979486 Break this problem into two subproblems by setting (f + 5) equal to 4.898979486 and -4.898979486.Subproblem 1
f + 5 = 4.898979486 Simplifying f + 5 = 4.898979486 Reorder the terms: 5 + f = 4.898979486 Solving 5 + f = 4.898979486 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '-5' to each side of the equation. 5 + -5 + f = 4.898979486 + -5 Combine like terms: 5 + -5 = 0 0 + f = 4.898979486 + -5 f = 4.898979486 + -5 Combine like terms: 4.898979486 + -5 = -0.101020514 f = -0.101020514 Simplifying f = -0.101020514Subproblem 2
f + 5 = -4.898979486 Simplifying f + 5 = -4.898979486 Reorder the terms: 5 + f = -4.898979486 Solving 5 + f = -4.898979486 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '-5' to each side of the equation. 5 + -5 + f = -4.898979486 + -5 Combine like terms: 5 + -5 = 0 0 + f = -4.898979486 + -5 f = -4.898979486 + -5 Combine like terms: -4.898979486 + -5 = -9.898979486 f = -9.898979486 Simplifying f = -9.898979486Solution
The solution to the problem is based on the solutions from the subproblems. f = {-0.101020514, -9.898979486}
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