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Simplifying f2 + 9f = -7 Reorder the terms: 9f + f2 = -7 Solving 9f + f2 = -7 Solving for variable 'f'. Reorder the terms: 7 + 9f + f2 = -7 + 7 Combine like terms: -7 + 7 = 0 7 + 9f + f2 = 0 Begin completing the square. Move the constant term to the right: Add '-7' to each side of the equation. 7 + 9f + -7 + f2 = 0 + -7 Reorder the terms: 7 + -7 + 9f + f2 = 0 + -7 Combine like terms: 7 + -7 = 0 0 + 9f + f2 = 0 + -7 9f + f2 = 0 + -7 Combine like terms: 0 + -7 = -7 9f + f2 = -7 The f term is 9f. Take half its coefficient (4.5). Square it (20.25) and add it to both sides. Add '20.25' to each side of the equation. 9f + 20.25 + f2 = -7 + 20.25 Reorder the terms: 20.25 + 9f + f2 = -7 + 20.25 Combine like terms: -7 + 20.25 = 13.25 20.25 + 9f + f2 = 13.25 Factor a perfect square on the left side: (f + 4.5)(f + 4.5) = 13.25 Calculate the square root of the right side: 3.640054945 Break this problem into two subproblems by setting (f + 4.5) equal to 3.640054945 and -3.640054945.Subproblem 1
f + 4.5 = 3.640054945 Simplifying f + 4.5 = 3.640054945 Reorder the terms: 4.5 + f = 3.640054945 Solving 4.5 + f = 3.640054945 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '-4.5' to each side of the equation. 4.5 + -4.5 + f = 3.640054945 + -4.5 Combine like terms: 4.5 + -4.5 = 0.0 0.0 + f = 3.640054945 + -4.5 f = 3.640054945 + -4.5 Combine like terms: 3.640054945 + -4.5 = -0.859945055 f = -0.859945055 Simplifying f = -0.859945055Subproblem 2
f + 4.5 = -3.640054945 Simplifying f + 4.5 = -3.640054945 Reorder the terms: 4.5 + f = -3.640054945 Solving 4.5 + f = -3.640054945 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '-4.5' to each side of the equation. 4.5 + -4.5 + f = -3.640054945 + -4.5 Combine like terms: 4.5 + -4.5 = 0.0 0.0 + f = -3.640054945 + -4.5 f = -3.640054945 + -4.5 Combine like terms: -3.640054945 + -4.5 = -8.140054945 f = -8.140054945 Simplifying f = -8.140054945Solution
The solution to the problem is based on the solutions from the subproblems. f = {-0.859945055, -8.140054945}
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