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Simplifying 3f2 + 2f = 6 Reorder the terms: 2f + 3f2 = 6 Solving 2f + 3f2 = 6 Solving for variable 'f'. Reorder the terms: -6 + 2f + 3f2 = 6 + -6 Combine like terms: 6 + -6 = 0 -6 + 2f + 3f2 = 0 Begin completing the square. Divide all terms by 3 the coefficient of the squared term: Divide each side by '3'. -2 + 0.6666666667f + f2 = 0 Move the constant term to the right: Add '2' to each side of the equation. -2 + 0.6666666667f + 2 + f2 = 0 + 2 Reorder the terms: -2 + 2 + 0.6666666667f + f2 = 0 + 2 Combine like terms: -2 + 2 = 0 0 + 0.6666666667f + f2 = 0 + 2 0.6666666667f + f2 = 0 + 2 Combine like terms: 0 + 2 = 2 0.6666666667f + f2 = 2 The f term is 0.6666666667f. Take half its coefficient (0.3333333334). Square it (0.1111111112) and add it to both sides. Add '0.1111111112' to each side of the equation. 0.6666666667f + 0.1111111112 + f2 = 2 + 0.1111111112 Reorder the terms: 0.1111111112 + 0.6666666667f + f2 = 2 + 0.1111111112 Combine like terms: 2 + 0.1111111112 = 2.1111111112 0.1111111112 + 0.6666666667f + f2 = 2.1111111112 Factor a perfect square on the left side: (f + 0.3333333334)(f + 0.3333333334) = 2.1111111112 Calculate the square root of the right side: 1.452966315 Break this problem into two subproblems by setting (f + 0.3333333334) equal to 1.452966315 and -1.452966315.Subproblem 1
f + 0.3333333334 = 1.452966315 Simplifying f + 0.3333333334 = 1.452966315 Reorder the terms: 0.3333333334 + f = 1.452966315 Solving 0.3333333334 + f = 1.452966315 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '-0.3333333334' to each side of the equation. 0.3333333334 + -0.3333333334 + f = 1.452966315 + -0.3333333334 Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000 0.0000000000 + f = 1.452966315 + -0.3333333334 f = 1.452966315 + -0.3333333334 Combine like terms: 1.452966315 + -0.3333333334 = 1.1196329816 f = 1.1196329816 Simplifying f = 1.1196329816Subproblem 2
f + 0.3333333334 = -1.452966315 Simplifying f + 0.3333333334 = -1.452966315 Reorder the terms: 0.3333333334 + f = -1.452966315 Solving 0.3333333334 + f = -1.452966315 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '-0.3333333334' to each side of the equation. 0.3333333334 + -0.3333333334 + f = -1.452966315 + -0.3333333334 Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000 0.0000000000 + f = -1.452966315 + -0.3333333334 f = -1.452966315 + -0.3333333334 Combine like terms: -1.452966315 + -0.3333333334 = -1.7862996484 f = -1.7862996484 Simplifying f = -1.7862996484Solution
The solution to the problem is based on the solutions from the subproblems. f = {1.1196329816, -1.7862996484}
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