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Simplifying 7f2 + f + -9 = 0 Reorder the terms: -9 + f + 7f2 = 0 Solving -9 + f + 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'. -1.285714286 + 0.1428571429f + f2 = 0 Move the constant term to the right: Add '1.285714286' to each side of the equation. -1.285714286 + 0.1428571429f + 1.285714286 + f2 = 0 + 1.285714286 Reorder the terms: -1.285714286 + 1.285714286 + 0.1428571429f + f2 = 0 + 1.285714286 Combine like terms: -1.285714286 + 1.285714286 = 0.000000000 0.000000000 + 0.1428571429f + f2 = 0 + 1.285714286 0.1428571429f + f2 = 0 + 1.285714286 Combine like terms: 0 + 1.285714286 = 1.285714286 0.1428571429f + f2 = 1.285714286 The f term is f. Take half its coefficient (0.5). Square it (0.25) and add it to both sides. Add '0.25' to each side of the equation. 0.1428571429f + 0.25 + f2 = 1.285714286 + 0.25 Reorder the terms: 0.25 + 0.1428571429f + f2 = 1.285714286 + 0.25 Combine like terms: 1.285714286 + 0.25 = 1.535714286 0.25 + 0.1428571429f + f2 = 1.535714286 Factor a perfect square on the left side: (f + 0.5)(f + 0.5) = 1.535714286 Calculate the square root of the right side: 1.239239398 Break this problem into two subproblems by setting (f + 0.5) equal to 1.239239398 and -1.239239398.Subproblem 1
f + 0.5 = 1.239239398 Simplifying f + 0.5 = 1.239239398 Reorder the terms: 0.5 + f = 1.239239398 Solving 0.5 + f = 1.239239398 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + f = 1.239239398 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + f = 1.239239398 + -0.5 f = 1.239239398 + -0.5 Combine like terms: 1.239239398 + -0.5 = 0.739239398 f = 0.739239398 Simplifying f = 0.739239398Subproblem 2
f + 0.5 = -1.239239398 Simplifying f + 0.5 = -1.239239398 Reorder the terms: 0.5 + f = -1.239239398 Solving 0.5 + f = -1.239239398 Solving for variable 'f'. Move all terms containing f to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + f = -1.239239398 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + f = -1.239239398 + -0.5 f = -1.239239398 + -0.5 Combine like terms: -1.239239398 + -0.5 = -1.739239398 f = -1.739239398 Simplifying f = -1.739239398Solution
The solution to the problem is based on the solutions from the subproblems. f = {0.739239398, -1.739239398}
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