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Simplifying f2 + -10f = 19 Reorder the terms: -10f + f2 = 19 Solving -10f + f2 = 19 Solving for variable 'f'. Reorder the terms: -19 + -10f + f2 = 19 + -19 Combine like terms: 19 + -19 = 0 -19 + -10f + f2 = 0 Begin completing the square. Move the constant term to the right: Add '19' to each side of the equation. -19 + -10f + 19 + f2 = 0 + 19 Reorder the terms: -19 + 19 + -10f + f2 = 0 + 19 Combine like terms: -19 + 19 = 0 0 + -10f + f2 = 0 + 19 -10f + f2 = 0 + 19 Combine like terms: 0 + 19 = 19 -10f + f2 = 19 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 = 19 + 25 Reorder the terms: 25 + -10f + f2 = 19 + 25 Combine like terms: 19 + 25 = 44 25 + -10f + f2 = 44 Factor a perfect square on the left side: (f + -5)(f + -5) = 44 Calculate the square root of the right side: 6.633249581 Break this problem into two subproblems by setting (f + -5) equal to 6.633249581 and -6.633249581.Subproblem 1
f + -5 = 6.633249581 Simplifying f + -5 = 6.633249581 Reorder the terms: -5 + f = 6.633249581 Solving -5 + f = 6.633249581 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 = 6.633249581 + 5 Combine like terms: -5 + 5 = 0 0 + f = 6.633249581 + 5 f = 6.633249581 + 5 Combine like terms: 6.633249581 + 5 = 11.633249581 f = 11.633249581 Simplifying f = 11.633249581Subproblem 2
f + -5 = -6.633249581 Simplifying f + -5 = -6.633249581 Reorder the terms: -5 + f = -6.633249581 Solving -5 + f = -6.633249581 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 = -6.633249581 + 5 Combine like terms: -5 + 5 = 0 0 + f = -6.633249581 + 5 f = -6.633249581 + 5 Combine like terms: -6.633249581 + 5 = -1.633249581 f = -1.633249581 Simplifying f = -1.633249581Solution
The solution to the problem is based on the solutions from the subproblems. f = {11.633249581, -1.633249581}
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