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Simplifying (-2f2 + -3f + -5) + (-2f2 + -3f + 8) = 0 Reorder the terms: (-5 + -3f + -2f2) + (-2f2 + -3f + 8) = 0 Remove parenthesis around (-5 + -3f + -2f2) -5 + -3f + -2f2 + (-2f2 + -3f + 8) = 0 Reorder the terms: -5 + -3f + -2f2 + (8 + -3f + -2f2) = 0 Remove parenthesis around (8 + -3f + -2f2) -5 + -3f + -2f2 + 8 + -3f + -2f2 = 0 Reorder the terms: -5 + 8 + -3f + -3f + -2f2 + -2f2 = 0 Combine like terms: -5 + 8 = 3 3 + -3f + -3f + -2f2 + -2f2 = 0 Combine like terms: -3f + -3f = -6f 3 + -6f + -2f2 + -2f2 = 0 Combine like terms: -2f2 + -2f2 = -4f2 3 + -6f + -4f2 = 0 Solving 3 + -6f + -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 + 1.5f + f2 = 0 Move the constant term to the right: Add '0.75' to each side of the equation. -0.75 + 1.5f + 0.75 + f2 = 0 + 0.75 Reorder the terms: -0.75 + 0.75 + 1.5f + f2 = 0 + 0.75 Combine like terms: -0.75 + 0.75 = 0.00 0.00 + 1.5f + f2 = 0 + 0.75 1.5f + f2 = 0 + 0.75 Combine like terms: 0 + 0.75 = 0.75 1.5f + f2 = 0.75 The f term is 1.5f. Take half its coefficient (0.75). Square it (0.5625) and add it to both sides. Add '0.5625' to each side of the equation. 1.5f + 0.5625 + f2 = 0.75 + 0.5625 Reorder the terms: 0.5625 + 1.5f + f2 = 0.75 + 0.5625 Combine like terms: 0.75 + 0.5625 = 1.3125 0.5625 + 1.5f + f2 = 1.3125 Factor a perfect square on the left side: (f + 0.75)(f + 0.75) = 1.3125 Calculate the square root of the right side: 1.145643924 Break this problem into two subproblems by setting (f + 0.75) equal to 1.145643924 and -1.145643924.Subproblem 1
f + 0.75 = 1.145643924 Simplifying f + 0.75 = 1.145643924 Reorder the terms: 0.75 + f = 1.145643924 Solving 0.75 + f = 1.145643924 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + f = 1.145643924 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + f = 1.145643924 + -0.75 f = 1.145643924 + -0.75 Combine like terms: 1.145643924 + -0.75 = 0.395643924 f = 0.395643924 Simplifying f = 0.395643924Subproblem 2
f + 0.75 = -1.145643924 Simplifying f + 0.75 = -1.145643924 Reorder the terms: 0.75 + f = -1.145643924 Solving 0.75 + f = -1.145643924 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '-0.75' to each side of the equation. 0.75 + -0.75 + f = -1.145643924 + -0.75 Combine like terms: 0.75 + -0.75 = 0.00 0.00 + f = -1.145643924 + -0.75 f = -1.145643924 + -0.75 Combine like terms: -1.145643924 + -0.75 = -1.895643924 f = -1.895643924 Simplifying f = -1.895643924Solution
The solution to the problem is based on the solutions from the subproblems. f = {0.395643924, -1.895643924}
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