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Simplifying f2 + 2f + -43 = 0 Reorder the terms: -43 + 2f + f2 = 0 Solving -43 + 2f + f2 = 0 Solving for variable 'f'. Begin completing the square. Move the constant term to the right: Add '43' to each side of the equation. -43 + 2f + 43 + f2 = 0 + 43 Reorder the terms: -43 + 43 + 2f + f2 = 0 + 43 Combine like terms: -43 + 43 = 0 0 + 2f + f2 = 0 + 43 2f + f2 = 0 + 43 Combine like terms: 0 + 43 = 43 2f + f2 = 43 The f term is 2f. Take half its coefficient (1). Square it (1) and add it to both sides. Add '1' to each side of the equation. 2f + 1 + f2 = 43 + 1 Reorder the terms: 1 + 2f + f2 = 43 + 1 Combine like terms: 43 + 1 = 44 1 + 2f + f2 = 44 Factor a perfect square on the left side: (f + 1)(f + 1) = 44 Calculate the square root of the right side: 6.633249581 Break this problem into two subproblems by setting (f + 1) equal to 6.633249581 and -6.633249581.Subproblem 1
f + 1 = 6.633249581 Simplifying f + 1 = 6.633249581 Reorder the terms: 1 + f = 6.633249581 Solving 1 + f = 6.633249581 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + f = 6.633249581 + -1 Combine like terms: 1 + -1 = 0 0 + f = 6.633249581 + -1 f = 6.633249581 + -1 Combine like terms: 6.633249581 + -1 = 5.633249581 f = 5.633249581 Simplifying f = 5.633249581Subproblem 2
f + 1 = -6.633249581 Simplifying f + 1 = -6.633249581 Reorder the terms: 1 + f = -6.633249581 Solving 1 + f = -6.633249581 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '-1' to each side of the equation. 1 + -1 + f = -6.633249581 + -1 Combine like terms: 1 + -1 = 0 0 + f = -6.633249581 + -1 f = -6.633249581 + -1 Combine like terms: -6.633249581 + -1 = -7.633249581 f = -7.633249581 Simplifying f = -7.633249581Solution
The solution to the problem is based on the solutions from the subproblems. f = {5.633249581, -7.633249581}
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