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Simplifying (3f2 + -1f + 7f) + (2f2 + -3f + -4) = 0 Reorder the terms: (-1f + 7f + 3f2) + (2f2 + -3f + -4) = 0 Combine like terms: -1f + 7f = 6f (6f + 3f2) + (2f2 + -3f + -4) = 0 Remove parenthesis around (6f + 3f2) 6f + 3f2 + (2f2 + -3f + -4) = 0 Reorder the terms: 6f + 3f2 + (-4 + -3f + 2f2) = 0 Remove parenthesis around (-4 + -3f + 2f2) 6f + 3f2 + -4 + -3f + 2f2 = 0 Reorder the terms: -4 + 6f + -3f + 3f2 + 2f2 = 0 Combine like terms: 6f + -3f = 3f -4 + 3f + 3f2 + 2f2 = 0 Combine like terms: 3f2 + 2f2 = 5f2 -4 + 3f + 5f2 = 0 Solving -4 + 3f + 5f2 = 0 Solving for variable 'f'. Begin completing the square. Divide all terms by 5 the coefficient of the squared term: Divide each side by '5'. -0.8 + 0.6f + f2 = 0 Move the constant term to the right: Add '0.8' to each side of the equation. -0.8 + 0.6f + 0.8 + f2 = 0 + 0.8 Reorder the terms: -0.8 + 0.8 + 0.6f + f2 = 0 + 0.8 Combine like terms: -0.8 + 0.8 = 0.0 0.0 + 0.6f + f2 = 0 + 0.8 0.6f + f2 = 0 + 0.8 Combine like terms: 0 + 0.8 = 0.8 0.6f + f2 = 0.8 The f term is 0.6f. Take half its coefficient (0.3). Square it (0.09) and add it to both sides. Add '0.09' to each side of the equation. 0.6f + 0.09 + f2 = 0.8 + 0.09 Reorder the terms: 0.09 + 0.6f + f2 = 0.8 + 0.09 Combine like terms: 0.8 + 0.09 = 0.89 0.09 + 0.6f + f2 = 0.89 Factor a perfect square on the left side: (f + 0.3)(f + 0.3) = 0.89 Calculate the square root of the right side: 0.943398113 Break this problem into two subproblems by setting (f + 0.3) equal to 0.943398113 and -0.943398113.Subproblem 1
f + 0.3 = 0.943398113 Simplifying f + 0.3 = 0.943398113 Reorder the terms: 0.3 + f = 0.943398113 Solving 0.3 + f = 0.943398113 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '-0.3' to each side of the equation. 0.3 + -0.3 + f = 0.943398113 + -0.3 Combine like terms: 0.3 + -0.3 = 0.0 0.0 + f = 0.943398113 + -0.3 f = 0.943398113 + -0.3 Combine like terms: 0.943398113 + -0.3 = 0.643398113 f = 0.643398113 Simplifying f = 0.643398113Subproblem 2
f + 0.3 = -0.943398113 Simplifying f + 0.3 = -0.943398113 Reorder the terms: 0.3 + f = -0.943398113 Solving 0.3 + f = -0.943398113 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '-0.3' to each side of the equation. 0.3 + -0.3 + f = -0.943398113 + -0.3 Combine like terms: 0.3 + -0.3 = 0.0 0.0 + f = -0.943398113 + -0.3 f = -0.943398113 + -0.3 Combine like terms: -0.943398113 + -0.3 = -1.243398113 f = -1.243398113 Simplifying f = -1.243398113Solution
The solution to the problem is based on the solutions from the subproblems. f = {0.643398113, -1.243398113}
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