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Simplifying 2f(3f + 2) = 3 Reorder the terms: 2f(2 + 3f) = 3 (2 * 2f + 3f * 2f) = 3 (4f + 6f2) = 3 Solving 4f + 6f2 = 3 Solving for variable 'f'. Reorder the terms: -3 + 4f + 6f2 = 3 + -3 Combine like terms: 3 + -3 = 0 -3 + 4f + 6f2 = 0 Begin completing the square. Divide all terms by 6 the coefficient of the squared term: Divide each side by '6'. -0.5 + 0.6666666667f + f2 = 0 Move the constant term to the right: Add '0.5' to each side of the equation. -0.5 + 0.6666666667f + 0.5 + f2 = 0 + 0.5 Reorder the terms: -0.5 + 0.5 + 0.6666666667f + f2 = 0 + 0.5 Combine like terms: -0.5 + 0.5 = 0.0 0.0 + 0.6666666667f + f2 = 0 + 0.5 0.6666666667f + f2 = 0 + 0.5 Combine like terms: 0 + 0.5 = 0.5 0.6666666667f + f2 = 0.5 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 = 0.5 + 0.1111111112 Reorder the terms: 0.1111111112 + 0.6666666667f + f2 = 0.5 + 0.1111111112 Combine like terms: 0.5 + 0.1111111112 = 0.6111111112 0.1111111112 + 0.6666666667f + f2 = 0.6111111112 Factor a perfect square on the left side: (f + 0.3333333334)(f + 0.3333333334) = 0.6111111112 Calculate the square root of the right side: 0.78173596 Break this problem into two subproblems by setting (f + 0.3333333334) equal to 0.78173596 and -0.78173596.Subproblem 1
f + 0.3333333334 = 0.78173596 Simplifying f + 0.3333333334 = 0.78173596 Reorder the terms: 0.3333333334 + f = 0.78173596 Solving 0.3333333334 + f = 0.78173596 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 = 0.78173596 + -0.3333333334 Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000 0.0000000000 + f = 0.78173596 + -0.3333333334 f = 0.78173596 + -0.3333333334 Combine like terms: 0.78173596 + -0.3333333334 = 0.4484026266 f = 0.4484026266 Simplifying f = 0.4484026266Subproblem 2
f + 0.3333333334 = -0.78173596 Simplifying f + 0.3333333334 = -0.78173596 Reorder the terms: 0.3333333334 + f = -0.78173596 Solving 0.3333333334 + f = -0.78173596 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 = -0.78173596 + -0.3333333334 Combine like terms: 0.3333333334 + -0.3333333334 = 0.0000000000 0.0000000000 + f = -0.78173596 + -0.3333333334 f = -0.78173596 + -0.3333333334 Combine like terms: -0.78173596 + -0.3333333334 = -1.1150692934 f = -1.1150692934 Simplifying f = -1.1150692934Solution
The solution to the problem is based on the solutions from the subproblems. f = {0.4484026266, -1.1150692934}
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