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Simplifying 7f2 + -4f + -6 = 0 Reorder the terms: -6 + -4f + 7f2 = 0 Solving -6 + -4f + 7f2 = 0 Solving for variable 'f'. Begin completing the square. Divide all terms by 7 the coefficient of the squared term: Divide each side by '7'. -0.8571428571 + -0.5714285714f + f2 = 0 Move the constant term to the right: Add '0.8571428571' to each side of the equation. -0.8571428571 + -0.5714285714f + 0.8571428571 + f2 = 0 + 0.8571428571 Reorder the terms: -0.8571428571 + 0.8571428571 + -0.5714285714f + f2 = 0 + 0.8571428571 Combine like terms: -0.8571428571 + 0.8571428571 = 0.0000000000 0.0000000000 + -0.5714285714f + f2 = 0 + 0.8571428571 -0.5714285714f + f2 = 0 + 0.8571428571 Combine like terms: 0 + 0.8571428571 = 0.8571428571 -0.5714285714f + f2 = 0.8571428571 The f term is -0.5714285714f. Take half its coefficient (-0.2857142857). Square it (0.08163265305) and add it to both sides. Add '0.08163265305' to each side of the equation. -0.5714285714f + 0.08163265305 + f2 = 0.8571428571 + 0.08163265305 Reorder the terms: 0.08163265305 + -0.5714285714f + f2 = 0.8571428571 + 0.08163265305 Combine like terms: 0.8571428571 + 0.08163265305 = 0.93877551015 0.08163265305 + -0.5714285714f + f2 = 0.93877551015 Factor a perfect square on the left side: (f + -0.2857142857)(f + -0.2857142857) = 0.93877551015 Calculate the square root of the right side: 0.968904283 Break this problem into two subproblems by setting (f + -0.2857142857) equal to 0.968904283 and -0.968904283.Subproblem 1
f + -0.2857142857 = 0.968904283 Simplifying f + -0.2857142857 = 0.968904283 Reorder the terms: -0.2857142857 + f = 0.968904283 Solving -0.2857142857 + f = 0.968904283 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '0.2857142857' to each side of the equation. -0.2857142857 + 0.2857142857 + f = 0.968904283 + 0.2857142857 Combine like terms: -0.2857142857 + 0.2857142857 = 0.0000000000 0.0000000000 + f = 0.968904283 + 0.2857142857 f = 0.968904283 + 0.2857142857 Combine like terms: 0.968904283 + 0.2857142857 = 1.2546185687 f = 1.2546185687 Simplifying f = 1.2546185687Subproblem 2
f + -0.2857142857 = -0.968904283 Simplifying f + -0.2857142857 = -0.968904283 Reorder the terms: -0.2857142857 + f = -0.968904283 Solving -0.2857142857 + f = -0.968904283 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '0.2857142857' to each side of the equation. -0.2857142857 + 0.2857142857 + f = -0.968904283 + 0.2857142857 Combine like terms: -0.2857142857 + 0.2857142857 = 0.0000000000 0.0000000000 + f = -0.968904283 + 0.2857142857 f = -0.968904283 + 0.2857142857 Combine like terms: -0.968904283 + 0.2857142857 = -0.6831899973 f = -0.6831899973 Simplifying f = -0.6831899973Solution
The solution to the problem is based on the solutions from the subproblems. f = {1.2546185687, -0.6831899973}
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