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Simplifying 3(4f + -1) * f = 2 Reorder the terms: 3(-1 + 4f) * f = 2 Reorder the terms for easier multiplication: 3f(-1 + 4f) = 2 (-1 * 3f + 4f * 3f) = 2 (-3f + 12f2) = 2 Solving -3f + 12f2 = 2 Solving for variable 'f'. Reorder the terms: -2 + -3f + 12f2 = 2 + -2 Combine like terms: 2 + -2 = 0 -2 + -3f + 12f2 = 0 Begin completing the square. Divide all terms by 12 the coefficient of the squared term: Divide each side by '12'. -0.1666666667 + -0.25f + f2 = 0 Move the constant term to the right: Add '0.1666666667' to each side of the equation. -0.1666666667 + -0.25f + 0.1666666667 + f2 = 0 + 0.1666666667 Reorder the terms: -0.1666666667 + 0.1666666667 + -0.25f + f2 = 0 + 0.1666666667 Combine like terms: -0.1666666667 + 0.1666666667 = 0.0000000000 0.0000000000 + -0.25f + f2 = 0 + 0.1666666667 -0.25f + f2 = 0 + 0.1666666667 Combine like terms: 0 + 0.1666666667 = 0.1666666667 -0.25f + f2 = 0.1666666667 The f term is -0.25f. Take half its coefficient (-0.125). Square it (0.015625) and add it to both sides. Add '0.015625' to each side of the equation. -0.25f + 0.015625 + f2 = 0.1666666667 + 0.015625 Reorder the terms: 0.015625 + -0.25f + f2 = 0.1666666667 + 0.015625 Combine like terms: 0.1666666667 + 0.015625 = 0.1822916667 0.015625 + -0.25f + f2 = 0.1822916667 Factor a perfect square on the left side: (f + -0.125)(f + -0.125) = 0.1822916667 Calculate the square root of the right side: 0.426956282 Break this problem into two subproblems by setting (f + -0.125) equal to 0.426956282 and -0.426956282.Subproblem 1
f + -0.125 = 0.426956282 Simplifying f + -0.125 = 0.426956282 Reorder the terms: -0.125 + f = 0.426956282 Solving -0.125 + f = 0.426956282 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '0.125' to each side of the equation. -0.125 + 0.125 + f = 0.426956282 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + f = 0.426956282 + 0.125 f = 0.426956282 + 0.125 Combine like terms: 0.426956282 + 0.125 = 0.551956282 f = 0.551956282 Simplifying f = 0.551956282Subproblem 2
f + -0.125 = -0.426956282 Simplifying f + -0.125 = -0.426956282 Reorder the terms: -0.125 + f = -0.426956282 Solving -0.125 + f = -0.426956282 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '0.125' to each side of the equation. -0.125 + 0.125 + f = -0.426956282 + 0.125 Combine like terms: -0.125 + 0.125 = 0.000 0.000 + f = -0.426956282 + 0.125 f = -0.426956282 + 0.125 Combine like terms: -0.426956282 + 0.125 = -0.301956282 f = -0.301956282 Simplifying f = -0.301956282Solution
The solution to the problem is based on the solutions from the subproblems. f = {0.551956282, -0.301956282}
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