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Simplifying 9f2 + -8f + -2 = 0 Reorder the terms: -2 + -8f + 9f2 = 0 Solving -2 + -8f + 9f2 = 0 Solving for variable 'f'. Begin completing the square. Divide all terms by 9 the coefficient of the squared term: Divide each side by '9'. -0.2222222222 + -0.8888888889f + f2 = 0 Move the constant term to the right: Add '0.2222222222' to each side of the equation. -0.2222222222 + -0.8888888889f + 0.2222222222 + f2 = 0 + 0.2222222222 Reorder the terms: -0.2222222222 + 0.2222222222 + -0.8888888889f + f2 = 0 + 0.2222222222 Combine like terms: -0.2222222222 + 0.2222222222 = 0.0000000000 0.0000000000 + -0.8888888889f + f2 = 0 + 0.2222222222 -0.8888888889f + f2 = 0 + 0.2222222222 Combine like terms: 0 + 0.2222222222 = 0.2222222222 -0.8888888889f + f2 = 0.2222222222 The f term is -0.8888888889f. Take half its coefficient (-0.4444444445). Square it (0.1975308642) and add it to both sides. Add '0.1975308642' to each side of the equation. -0.8888888889f + 0.1975308642 + f2 = 0.2222222222 + 0.1975308642 Reorder the terms: 0.1975308642 + -0.8888888889f + f2 = 0.2222222222 + 0.1975308642 Combine like terms: 0.2222222222 + 0.1975308642 = 0.4197530864 0.1975308642 + -0.8888888889f + f2 = 0.4197530864 Factor a perfect square on the left side: (f + -0.4444444445)(f + -0.4444444445) = 0.4197530864 Calculate the square root of the right side: 0.647883544 Break this problem into two subproblems by setting (f + -0.4444444445) equal to 0.647883544 and -0.647883544.Subproblem 1
f + -0.4444444445 = 0.647883544 Simplifying f + -0.4444444445 = 0.647883544 Reorder the terms: -0.4444444445 + f = 0.647883544 Solving -0.4444444445 + f = 0.647883544 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '0.4444444445' to each side of the equation. -0.4444444445 + 0.4444444445 + f = 0.647883544 + 0.4444444445 Combine like terms: -0.4444444445 + 0.4444444445 = 0.0000000000 0.0000000000 + f = 0.647883544 + 0.4444444445 f = 0.647883544 + 0.4444444445 Combine like terms: 0.647883544 + 0.4444444445 = 1.0923279885 f = 1.0923279885 Simplifying f = 1.0923279885Subproblem 2
f + -0.4444444445 = -0.647883544 Simplifying f + -0.4444444445 = -0.647883544 Reorder the terms: -0.4444444445 + f = -0.647883544 Solving -0.4444444445 + f = -0.647883544 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '0.4444444445' to each side of the equation. -0.4444444445 + 0.4444444445 + f = -0.647883544 + 0.4444444445 Combine like terms: -0.4444444445 + 0.4444444445 = 0.0000000000 0.0000000000 + f = -0.647883544 + 0.4444444445 f = -0.647883544 + 0.4444444445 Combine like terms: -0.647883544 + 0.4444444445 = -0.2034390995 f = -0.2034390995 Simplifying f = -0.2034390995Solution
The solution to the problem is based on the solutions from the subproblems. f = {1.0923279885, -0.2034390995}
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