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Simplifying -3f2 + 5f + 6 = 0 Reorder the terms: 6 + 5f + -3f2 = 0 Solving 6 + 5f + -3f2 = 0 Solving for variable 'f'. Begin completing the square. Divide all terms by -3 the coefficient of the squared term: Divide each side by '-3'. -2 + -1.666666667f + f2 = 0 Move the constant term to the right: Add '2' to each side of the equation. -2 + -1.666666667f + 2 + f2 = 0 + 2 Reorder the terms: -2 + 2 + -1.666666667f + f2 = 0 + 2 Combine like terms: -2 + 2 = 0 0 + -1.666666667f + f2 = 0 + 2 -1.666666667f + f2 = 0 + 2 Combine like terms: 0 + 2 = 2 -1.666666667f + f2 = 2 The f term is -1.666666667f. Take half its coefficient (-0.8333333335). Square it (0.6944444447) and add it to both sides. Add '0.6944444447' to each side of the equation. -1.666666667f + 0.6944444447 + f2 = 2 + 0.6944444447 Reorder the terms: 0.6944444447 + -1.666666667f + f2 = 2 + 0.6944444447 Combine like terms: 2 + 0.6944444447 = 2.6944444447 0.6944444447 + -1.666666667f + f2 = 2.6944444447 Factor a perfect square on the left side: (f + -0.8333333335)(f + -0.8333333335) = 2.6944444447 Calculate the square root of the right side: 1.6414763 Break this problem into two subproblems by setting (f + -0.8333333335) equal to 1.6414763 and -1.6414763.Subproblem 1
f + -0.8333333335 = 1.6414763 Simplifying f + -0.8333333335 = 1.6414763 Reorder the terms: -0.8333333335 + f = 1.6414763 Solving -0.8333333335 + f = 1.6414763 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '0.8333333335' to each side of the equation. -0.8333333335 + 0.8333333335 + f = 1.6414763 + 0.8333333335 Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000 0.0000000000 + f = 1.6414763 + 0.8333333335 f = 1.6414763 + 0.8333333335 Combine like terms: 1.6414763 + 0.8333333335 = 2.4748096335 f = 2.4748096335 Simplifying f = 2.4748096335Subproblem 2
f + -0.8333333335 = -1.6414763 Simplifying f + -0.8333333335 = -1.6414763 Reorder the terms: -0.8333333335 + f = -1.6414763 Solving -0.8333333335 + f = -1.6414763 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '0.8333333335' to each side of the equation. -0.8333333335 + 0.8333333335 + f = -1.6414763 + 0.8333333335 Combine like terms: -0.8333333335 + 0.8333333335 = 0.0000000000 0.0000000000 + f = -1.6414763 + 0.8333333335 f = -1.6414763 + 0.8333333335 Combine like terms: -1.6414763 + 0.8333333335 = -0.8081429665 f = -0.8081429665 Simplifying f = -0.8081429665Solution
The solution to the problem is based on the solutions from the subproblems. f = {2.4748096335, -0.8081429665}
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