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Simplifying 4f2 + 3f + -3 = 0 Reorder the terms: -3 + 3f + 4f2 = 0 Solving -3 + 3f + 4f2 = 0 Solving for variable 'f'. Begin completing the square. Divide all terms by 4 the coefficient of the squared term: Divide each side by '4'. -0.75 + 0.75f + f2 = 0 Move the constant term to the right: Add '0.75' to each side of the equation. -0.75 + 0.75f + 0.75 + f2 = 0 + 0.75 Reorder the terms: -0.75 + 0.75 + 0.75f + f2 = 0 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + 0.75f + f2 = 0 + 0.75 0.75f + f2 = 0 + 0.75 Combine like terms: 0 + 0.75 = 0.75 0.75f + f2 = 0.75 The f term is 0.75f. Take half its coefficient (0.375). Square it (0.140625) and add it to both sides. Add '0.140625' to each side of the equation. 0.75f + 0.140625 + f2 = 0.75 + 0.140625 Reorder the terms: 0.140625 + 0.75f + f2 = 0.75 + 0.140625 Combine like terms: 0.75 + 0.140625 = 0.890625 0.140625 + 0.75f + f2 = 0.890625 Factor a perfect square on the left side: (f + 0.375)(f + 0.375) = 0.890625 Calculate the square root of the right side: 0.943729304 Break this problem into two subproblems by setting (f + 0.375) equal to 0.943729304 and -0.943729304.Subproblem 1
f + 0.375 = 0.943729304 Simplifying f + 0.375 = 0.943729304 Reorder the terms: 0.375 + f = 0.943729304 Solving 0.375 + f = 0.943729304 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '-0.375' to each side of the equation. 0.375 + -0.375 + f = 0.943729304 + -0.375 Combine like terms: 0.375 + -0.375 = 0.000 0.000 + f = 0.943729304 + -0.375 f = 0.943729304 + -0.375 Combine like terms: 0.943729304 + -0.375 = 0.568729304 f = 0.568729304 Simplifying f = 0.568729304Subproblem 2
f + 0.375 = -0.943729304 Simplifying f + 0.375 = -0.943729304 Reorder the terms: 0.375 + f = -0.943729304 Solving 0.375 + f = -0.943729304 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '-0.375' to each side of the equation. 0.375 + -0.375 + f = -0.943729304 + -0.375 Combine like terms: 0.375 + -0.375 = 0.000 0.000 + f = -0.943729304 + -0.375 f = -0.943729304 + -0.375 Combine like terms: -0.943729304 + -0.375 = -1.318729304 f = -1.318729304 Simplifying f = -1.318729304Solution
The solution to the problem is based on the solutions from the subproblems. f = {0.568729304, -1.318729304}
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