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Simplifying f2 + 2f = 7 Reorder the terms: 2f + f2 = 7 Solving 2f + f2 = 7 Solving for variable 'f'. Reorder the terms: -7 + 2f + f2 = 7 + -7 Combine like terms: 7 + -7 = 0 -7 + 2f + f2 = 0 Begin completing the square. Move the constant term to the right: Add '7' to each side of the equation. -7 + 2f + 7 + f2 = 0 + 7 Reorder the terms: -7 + 7 + 2f + f2 = 0 + 7 Combine like terms: -7 + 7 = 0 0 + 2f + f2 = 0 + 7 2f + f2 = 0 + 7 Combine like terms: 0 + 7 = 7 2f + f2 = 7 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 = 7 + 1 Reorder the terms: 1 + 2f + f2 = 7 + 1 Combine like terms: 7 + 1 = 8 1 + 2f + f2 = 8 Factor a perfect square on the left side: (f + 1)(f + 1) = 8 Calculate the square root of the right side: 2.828427125 Break this problem into two subproblems by setting (f + 1) equal to 2.828427125 and -2.828427125.Subproblem 1
f + 1 = 2.828427125 Simplifying f + 1 = 2.828427125 Reorder the terms: 1 + f = 2.828427125 Solving 1 + f = 2.828427125 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 = 2.828427125 + -1 Combine like terms: 1 + -1 = 0 0 + f = 2.828427125 + -1 f = 2.828427125 + -1 Combine like terms: 2.828427125 + -1 = 1.828427125 f = 1.828427125 Simplifying f = 1.828427125Subproblem 2
f + 1 = -2.828427125 Simplifying f + 1 = -2.828427125 Reorder the terms: 1 + f = -2.828427125 Solving 1 + f = -2.828427125 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 = -2.828427125 + -1 Combine like terms: 1 + -1 = 0 0 + f = -2.828427125 + -1 f = -2.828427125 + -1 Combine like terms: -2.828427125 + -1 = -3.828427125 f = -3.828427125 Simplifying f = -3.828427125Solution
The solution to the problem is based on the solutions from the subproblems. f = {1.828427125, -3.828427125}
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