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Simplifying f2 + 12f = -29 Reorder the terms: 12f + f2 = -29 Solving 12f + f2 = -29 Solving for variable 'f'. Reorder the terms: 29 + 12f + f2 = -29 + 29 Combine like terms: -29 + 29 = 0 29 + 12f + f2 = 0 Begin completing the square. Move the constant term to the right: Add '-29' to each side of the equation. 29 + 12f + -29 + f2 = 0 + -29 Reorder the terms: 29 + -29 + 12f + f2 = 0 + -29 Combine like terms: 29 + -29 = 0 0 + 12f + f2 = 0 + -29 12f + f2 = 0 + -29 Combine like terms: 0 + -29 = -29 12f + f2 = -29 The f term is 12f. Take half its coefficient (6). Square it (36) and add it to both sides. Add '36' to each side of the equation. 12f + 36 + f2 = -29 + 36 Reorder the terms: 36 + 12f + f2 = -29 + 36 Combine like terms: -29 + 36 = 7 36 + 12f + f2 = 7 Factor a perfect square on the left side: (f + 6)(f + 6) = 7 Calculate the square root of the right side: 2.645751311 Break this problem into two subproblems by setting (f + 6) equal to 2.645751311 and -2.645751311.Subproblem 1
f + 6 = 2.645751311 Simplifying f + 6 = 2.645751311 Reorder the terms: 6 + f = 2.645751311 Solving 6 + f = 2.645751311 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '-6' to each side of the equation. 6 + -6 + f = 2.645751311 + -6 Combine like terms: 6 + -6 = 0 0 + f = 2.645751311 + -6 f = 2.645751311 + -6 Combine like terms: 2.645751311 + -6 = -3.354248689 f = -3.354248689 Simplifying f = -3.354248689Subproblem 2
f + 6 = -2.645751311 Simplifying f + 6 = -2.645751311 Reorder the terms: 6 + f = -2.645751311 Solving 6 + f = -2.645751311 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '-6' to each side of the equation. 6 + -6 + f = -2.645751311 + -6 Combine like terms: 6 + -6 = 0 0 + f = -2.645751311 + -6 f = -2.645751311 + -6 Combine like terms: -2.645751311 + -6 = -8.645751311 f = -8.645751311 Simplifying f = -8.645751311Solution
The solution to the problem is based on the solutions from the subproblems. f = {-3.354248689, -8.645751311}
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