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Simplifying 5f2 + 7f + -4 = 0 Reorder the terms: -4 + 7f + 5f2 = 0 Solving -4 + 7f + 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 + 1.4f + f2 = 0 Move the constant term to the right: Add '0.8' to each side of the equation. -0.8 + 1.4f + 0.8 + f2 = 0 + 0.8 Reorder the terms: -0.8 + 0.8 + 1.4f + f2 = 0 + 0.8 Combine like terms: -0.8 + 0.8 = 0.0 0.0 + 1.4f + f2 = 0 + 0.8 1.4f + f2 = 0 + 0.8 Combine like terms: 0 + 0.8 = 0.8 1.4f + f2 = 0.8 The f term is 1.4f. Take half its coefficient (0.7). Square it (0.49) and add it to both sides. Add '0.49' to each side of the equation. 1.4f + 0.49 + f2 = 0.8 + 0.49 Reorder the terms: 0.49 + 1.4f + f2 = 0.8 + 0.49 Combine like terms: 0.8 + 0.49 = 1.29 0.49 + 1.4f + f2 = 1.29 Factor a perfect square on the left side: (f + 0.7)(f + 0.7) = 1.29 Calculate the square root of the right side: 1.135781669 Break this problem into two subproblems by setting (f + 0.7) equal to 1.135781669 and -1.135781669.Subproblem 1
f + 0.7 = 1.135781669 Simplifying f + 0.7 = 1.135781669 Reorder the terms: 0.7 + f = 1.135781669 Solving 0.7 + f = 1.135781669 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '-0.7' to each side of the equation. 0.7 + -0.7 + f = 1.135781669 + -0.7 Combine like terms: 0.7 + -0.7 = 0.0 0.0 + f = 1.135781669 + -0.7 f = 1.135781669 + -0.7 Combine like terms: 1.135781669 + -0.7 = 0.435781669 f = 0.435781669 Simplifying f = 0.435781669Subproblem 2
f + 0.7 = -1.135781669 Simplifying f + 0.7 = -1.135781669 Reorder the terms: 0.7 + f = -1.135781669 Solving 0.7 + f = -1.135781669 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '-0.7' to each side of the equation. 0.7 + -0.7 + f = -1.135781669 + -0.7 Combine like terms: 0.7 + -0.7 = 0.0 0.0 + f = -1.135781669 + -0.7 f = -1.135781669 + -0.7 Combine like terms: -1.135781669 + -0.7 = -1.835781669 f = -1.835781669 Simplifying f = -1.835781669Solution
The solution to the problem is based on the solutions from the subproblems. f = {0.435781669, -1.835781669}
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